An Introduction to the Linear Representations of Finite
Groups
Rafik Ballou
To cite this version:
Rafik Ballou. An Introduction to the Linear Representations of Finite Groups. EDP Sciences.
Contribution of Symmetries in Condensed Matter, EPJ Web of Conferences, pp.00005, 2012,
EPJ Web of Conferences, <10.1051/epjconf/20122200005>. <hal-00963924>
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EPJ Web of Conferences 22, 00005 (2012)
DOI: 10.1051/epjconf/20122200005
C Owned by the authors, published by EDP Sciences, 2012
An introduction to the linear representations of finite groups
R. Ballou
Institut Néel, CNRS / UJF, 25 rue des Martyrs, BP. 166, 38042 Grenoble Cedex 9, France
Abstract. A few elements of the formalism of finite group representations are recalled. As to avoid a too
mathematically oriented approach the discussed items are limited to the most essential aspects of the linear
and matrix representations of standard use in chemistry and physics.
1. INTRODUCTION
Symmetry is ubiquitous in nature and of an extremely wide variety. It may be discrete, such as the space
inversion, the time inversion, the crystal isometries, . . . , or continuous, such as the euclidean isometries,
the galilean invariance, the gauge invariances, . . . . It may be obvious, generally when it is of geometric
nature. It may be hidden, often when it is of dynamical origin, then revealing itself indirectly.1 It may be
more or less blurred, typically as perceived in the complex systems, botanical, biological, . . . . It may be
spontaneously broken, in which instance it becomes the source of a number of non-trivial phenomena,
including the phase transitions, the bifurcations in the non linear processes, . . . , that gives rise to a
wealth of structuration. It suffices for illustration to evoke the uncountable physical phases of matter,
for instance the crystalline and mesomorphic forms or else the magnetic orders among the most familiar
categories, not to mention the dynamic self-organization, the pattern formation, . . . found out in the
other fields. Symmetry in fact scarcely is lowered uniformly so that the broken phase spatially builds up
from different states, transforming into one another by the lost components of the symmetry, and thus is
non uniform and displays defects. These in turn might interact or cross, possibly non commutatively, to
organize themselves or generate further novel textures.
Symmetry gets materialized through a set of transformations of the properties of a system, which,
endowed with the canonical composition law for functions, forms a group whatever the instance.
Accordingly, the adequate framework within which to deal with symmetry is that of the group theory,
including its ramifications into the representation theory to account for the nature of the invariances
of the physical properties, the differential geometry, in particular the Morse theory, to investigate the
extrema of the invariant functions of the physical properties and thus to get insights into the symmetry
breaking phenomena, the algebraic topology, more specifically the homotopy theory, to feature the
topological stability of defects and the formation of textures, . . . . It is clear that this is too vast a field to
1 A case in point is provided by the bound states of the non relativistic isolated hydrogen atom, which displays spectral
degeneracies with respect to the principal n and orbital l quantum numbers. Whereas the l-degeneracy is an evident outcome
of the symmetry group SO(3) of the rotations in the 3-dimensional space R3 , the n-degeneracy is specific to the Kepler potentials,
decreasing as the inverse of the radial distance, and emanates from the dynamical symmetry group SO(4). Considering the
scattering states of the continuum in the spectrum, this metamorphoses itself into the dynamical symmetry group SO(3,1). In
other words, using a more intuitive picture, the electron dynamics in a 1/r potential is equivalent to that of a free particle in the
4-dimensional space R4 , on a sphere S3 if it is bounded and on a double-sheeted hyperboloid H3 if it is scattered. Another feature
of the electron spectrum is the equal spacing of the energy levels when multiplied by −n3 , which suggests duality and originates
from the De Sitter spectrum generating symmetry group SO(4,1). Attempts to express the hamiltonian in terms of operators that
close under commutation lead to anticipate that the largest spectrum generating symmetry group of the hydrogen atom might be
the conformal group SO(4,2).
This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which
permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20122200005
EPJ Web of Conferences
describe in a few pages. The ambition of these notes is strongly limited. It is to focus on the mathematics
of the linear representation of finite groups. After short recalls of basic concepts, questions of reduction
and irreducibility are discussed. Next, character theory is succinctly explored. Complete reducibility of
the linear representations of finite groups, the relevance and usefulness of the Schur’s Lemmas, complete
invariant nature of the characters with respect to intertwining and character completeness over class
functions are emphasized. Construction of induced linear representations will be approached and search
methods of irreducible representations will be mentioned only briefly. Of course, the discussed items
are far from providing even the rough idea of all the richness of the group representations. A number
of their facets are only alluded to or merely ignored, for instance concerned with the multi-valued
spinor representations, the projective representations, . . . , not to mention the linear representations of
continuous groups or else the non linear group actions. An extremely wide literature exists on these
topics, quite often purely mathematical, including textbooks or reviews to start with. See for instance
[1–5].
2. BASIC CONCEPTS
A representation of a group G on a mathematical object X designates an homomorphism : G →
Aut(X) from the group G to the automorphism group Aut(X) of the object X:
(gh) = (g) ◦ (h)
∀g ∈ G ∀h ∈ G
(2.1)
G may be any group, finite or infinite, possibly topological in which case it may be (locally) compact or
non compact, n-connected, . . . . X may be any set endowed with a mathematical structure, for instance
a topological space, a differentiable manifold, a module over a ring, . . . . Aut(X) is the group formed by
the set of the bijective functions f : X → X that preserve the mathematical structure of X, endowed with
the canonical composition law ◦ for the functions.
If X is a vector space V over a scalar field K then Aut(V) is the group GL(V, K) of the invertible
linear operators on V:
(g)(a u
+ b v ) = a (g)(
u) + b (g)(v )
u, v ) ∈ V2
∀g ∈ G ∀(a, b) ∈ K2 ∀(
(2.2)
In this case is particularized by naming it a linear representation. V is the representation space. It is
customary to call dimension of the representation the dimension d of V. Only the linear representations
of the finite groups G on the vector spaces V over the field C of the complex numbers2 are discussed
in this manuscript, unless otherwise explicitly stated.
With every linear representation : G → GL(V, K) is associated its kernel ker() and its image
im(), given as
ker() = {g ∈ G | (g) = 1V } and
im() = {(g) | g ∈ G}
(2.3)
where 1V ∈ GL(V, K) is the identity operator on the representation space V. If (g) = (h) then
gh−1 ∈ ker(). It follows that is injective if and only if (iff) ker() = {e}, where e is the unit element
of G. by definition is surjective iff im() = GL(V, K). If (g, h) ∈ ker()2 then (gh−1 ) = 1V , namely
gh−1 ∈ ker(), which implies that ker() is a subgroup of G. It similarly is shown that im() is a
subgroup of GL(V, K). If g ∈ G and h ∈ ker() then (ghg −1 ) = 1V , namely ghg −1 ∈ ker(), which
2 An evident motivation to restrict the scalar field K to the field C of the complex numbers is of course that physics suggests it
as natural, together with the field R of the real numbers. A mathematical motivation is that this avoids unnecessarily cautioning
against a number of algebraic stuffs, because C has zero characteristic, similarly as R, and is algebraically closed, in contrast
to R. These two mathematical properties are relevant to certain aspects of crucial theorems, such as the Complete Reducibility
Theorem, the Schur’s Lemma, . . . . The characteristic char(K) of a field K is the positive integer nK the multiples of which makes
up the kernel nK Z of the homomorphism : m → 1K + . . . + 1K = m · 1K of the additive group of the integer numbers Z to the
additive group of the K-scalars. char(K) by convention is set to zero whenever nK is not finite. A field is algebraically closed if
for every polynomial P(z) of one variable and coefficients in this field, ∃z 0 s.t. P(z 0 ) = 0.
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Contribution of Symmetries in Condensed Matter
means that ker() is a normal subgroup of G,
ker() G
(2.4)
A quotient group G/ker() is canonically defined by endowing the set of the left cosets of ker()
with the internal law of composition : (g ker(), h ker()) → (gh) ker(). It almost is obvious that
G/ker() is isomorphic to im(),
G/ker() ∼
= im()
(2.5)
All these properties actually are generic to the kernel and image of any homomorphism of groups.
A linear representation : G → GL(V, K) is isomorphic by definition to a linear representation
: G → GL(W, K) if there exists an isomorphism : V → W, which is equivariant:
(g) ◦ = ◦ (g)
∀g ∈ G
(2.6)
is called an intertwining operator or a G-linear map. A standard notation for isomorphic
representations is ∼ . The isomorphism of representations is reflexive, symmetric and transitive,
so is an equivalence relation, which gathers the linear representations into equivalence classes.
Any vector space V possesses a dual V# , which canonically is built up by endowing the set of linear
=u
# (w)
+ v # (w)
and pointwise scalar multiplication
forms on V with pointwise addition (
u# + v # )(w)
#
#
#
#
( v )(w)
= v (w),
where ∈ C, w
∈ V, u
∈ V and v # ∈ V# .3 Let : G → GL(V, C) be a linear
representation. An application # : G → GL(V# , C), g → # (g) such that
#
(g)(v # ) ((g)(
u)) = v # (
u) ∀g ∈ G, ∀
u ∈ V, ∀v # ∈ V#
(2.7)
#
can be defined. With w
= (g)(
u), this is rewritten (g)(v # ) (w)
= v # (g)−1 (w)
. In other words
# (g)(v # ) = v # ◦ (g −1 ), which makes up another equivalent defining relation and clearly shows
that # (g) does exist and is unique thanks to the existence and unicity of (g −1 ) ∀g ∈ G. Moreover, # (gh)(v # ) = v # ◦ ((gh)−1 ) = v # ◦ (h−1 ) ◦ (g −1 ) = # (g)(v # ◦ (h−1 )) = # (g)(# (h)(v # )) =
(# (g) ◦ # (h))(v # ), ∀v # ∈ V# , ∀(g, h) ∈ G2 , which demonstrates that # is a group homomorphism.
# is the dual representation of . All the theorems established for are valid for # , and conversely,
by mere structure transport.
2.1 Canonical examples
Automorphism groups GL(Vd=1 , C) of 1-dimensional vector spaces Vd=1 are isomorphic to the
multiplicative group C⋆ of non null complex numbers, insofar as every invertible linear operator on Vd=1 is equivalent to the multiplication by a same non null scalar: if eˆ is the basis vector in
Vd=1 then ∃!a ∈ C⋆ s.t. (ˆe) = a eˆ ergo ∀v ∈ Vd=1 (v ) = (v eˆ ) = v(ˆe) = va eˆ = av eˆ = av . Any
homomorphism : G → C⋆ thus makes up a linear representation of dimension d = 1 of the
group G. An evocative example is : GL(d, C) → C⋆ , M → Det(M), where GL(d, C) designates
the group of d × d non singular matrices with entries in C and Det(M) the determinant of a matrix
M. ker() = SL(d, C) consists in the d × d matrices with determinant 1, which thus is a normal
subgroup of GL(d, C). Since im() = C⋆ we have GL(d, C)/SL(d, C) ∼
= C⋆ . GL(d, C) is called the
general linear group of order d over C and SL(d, C) the special linear group of order d over C. It
3 A linear form by definition is an application v
# : V → C from a vector space V to its scalar field C such that v # (a r + b s ) =
a v # (r ) + b v # (s ), ∀(a, b) ∈ C2 , ∀(r , s ) ∈ V2 . It also is called a one-form, a linear functional, a co-vector, a contravariant vector
when the elements of V are called covariant vectors, . . . . This merely emphasizes the wealth of context within which the concept
might be in use, such as differential geometry, measure theory, multilinear algebra, . . . . If V has the finite dimension d then
V# has the same dimension d. A basis {eˆi # }i=1,...,d in V# is twinned in fact to any selected basis {eˆi }i=1,...,d in V such that
eˆi # (eˆj ) = ij , where ij is the Kronecker symbol (ij = 1 iff i = j and ij = 0 otherwise). When V is infinite-dimensional the
same construction does not end up with a basis. It leads to a family of linearly independent vectors that is not spanning. The linear
forms on a finite-dimensional normed space V are bounded and therefore are continuous.
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EPJ Web of Conferences
is clear that g m ∈ G ⇒ g m+1 ∈ G ∀m ∈ N whence, by increasing to infinity, m crosses integers p
for which there exists strictly positive integers q < p such that g p = g q or else g p−q = e, unless G
is infinite. In other words, whenever the group G is finite each of its element g is of finite order
ng = Minimum [r ∈ N⋆ | g r = e]. Obviously, (g)ng = (g ng ) = (e) = 1, which means that (g) is
an ng --th root of 1 , the multiplicative unit of C⋆ . Now whatever the group G, finite or not, in the event
that one has
(g) = 1 ∀g ∈ G
(2.8)
is called the trivial representation of the group G. Its significance is to reveal the full invariance of a
physical property with respect to the symmetries abstracted by the elements of the group G.
Indexing with the elements x of a finite set X the basis vectors eˆ x of a vector space V and associating
each element g of a finite group G with the invertible linear operator X (g) on V that sends eˆ x to eˆ (g)(x) ,
where : G → PX is an homomorphism of the group G into the group PX of the permutations of X,
generates a linear representation X , which is called the permutation representation of the group G
associated with the set X. Note that the group homomorphism : G → PX defines a representation of
the group G on the set X. It is the usage in that case to state that the group G acts on the set X or else
that X is a G-set. In the specific instance where the set X contains the same number nG of elements as
the group G the permutation representation is isomorphic to the so-called regular representation G of
the group G. One conventionally defines G by indexing the basis vectors of the vector space V with the
elements h of the group G, more concisely as eˆ h where h ∈ G, and by associating each element g of the
group G with the invertible linear operator G (g) on V that transforms the basis vectors, thus G-indexed,
according to the formula
G (g)(ˆeh ) = eˆ gh
∀g ∈ G ∀h ∈ G
(2.9)
The regular representation G is particularized because containing each irreducible representation i
of the group G with a repetition factor equal to its dimension di . The dimension of G is the order
nG of the group G. The set {G (g)(ˆee ) | g ∈ G}, engendered from the single vector eˆ e indexed with
the unit element e of the group G, forms a basis of the representation space V. Conversely, given a
linear representation : G → GL(V, C), if there exist a vector v in the representation space V such
that the set {(g)(v ) | g ∈ G} forms a basis of V then necessarily is isomorphic to G . Consider
indeed the isomorphism : V → V defined by setting (ˆeh ) = (h)v . Since is an homomorphism,
∀(g, h) ∈ G2 , (g) ((h)(v )) = (gh)(v ), but, by definition of , (h)(v ) = (ˆeh ) and (gh)(v ) = (ˆegh )
so that (g) ((ˆeh )) = (G (g)(ˆeh )), which implies that (g) ◦ = ◦ G (g) ∀g ∈ G, namely that is
equivariant, whence ∼ G .
2.2 Matrix representations
Let V be a vector space with dimension d over the field C. Any element (g) of the group GL(V,C)
of the invertible linear operators on V is fully determined from the images (g)(ˆ
em ) of the basis
(m = 1, . . . , d) selected in V. Indeed, ∀v ∈ V ∃! (x1 , . . . , xd ) ∈ Cd : v = m xm eˆ m so that
vectors eˆ m (g)(v ) = m xm (g)(ˆem ). Now, (g)(ˆem ) is a vector of V. Accordingly, ∃! ((g)1m , . . . , (g)dm ) ∈ Cd
such that
(g)(ˆem ) =
eˆ n (g)nm
(2.10)
n
If (g)(v ) = n yn eˆ n then yn = m (g)nm xm . The d 2 complex coefficients (g)nm make up the
entries of a d × d invertible matrix (g), called the matrix representative of the linear operator
(g). Assume that (g) generically symbolizes the image of an element g of a group G by a
linear representation : G → GL(V, C), so that ∀ g, h ∈ G, (gh) = (g) ◦ (h). It follows from
00005-p.4
Contribution of Symmetries in Condensed Matter
the
equation
(2.10) that (gh)(ˆem ) = n eˆ n (gh)nm and (g) ◦ (h)(ˆem ) = (g)( s eˆ s (h)sm ) =
ˆ n [ s (g)ns (h)sm ]. Accordingly,
ne
(gh) = (g)(h)
∀g ∈ G ∀h ∈ G
(2.11)
This means that the mapping : G → GL(d, C) of the group G to the group GL(d, C) of d × d invertible
matrices with entries in C, which to each element g in G associates the matrix representative (g) of the
linear operator (g) with respect to the selected basis {ˆe}m=1,...,d , defines a group homomorphism. This
is called a matrix representation of the group G.
The selection of another basis {fˆ }n=1,...,d would have led to other matrix representatives (g), giving
rise to another matrix representation : G → GL(d, C). Associated with the same linear representation
and merely emerging from the selection of two different bases in the representation space V, the matrix
representations and are said similar or equivalent. If S is the invertible matrix associated with the
basis change {ˆe}m=1,...,d → {fˆ }n=1,...,d , which often is called a similarity transformation, then4
(g) = S (g) S −1
∀g ∈ G
(2.12)
and are said intertwined with S. Conversely, any two finite dimensional matrix representations
of a finite group intertwined with an invertible matrix are similar. As with the linear representations,
a standard notation for two equivalent matrix representations is ∼ . Now, (g) could have
been interpreted also as the matrix representative with respect to the initial basis vectors eˆ m (m =
1, . . . , d) of a linear operator (g) associated with another linear representation : G → GL(V, C). The
equation (2.12) then would mean that there exists an automorphism of V which is equivariant: (g) =
◦ (g) ◦ −1 , ∀g ∈ G so that ∼ . Conversely, any automorphism of V corresponds to a change
of bases. Accordingly, the isomorphism of linear representations and that of matrix representations
describe the same equivalence.
As from every matrix M with entries Mij in C is built the complex conjugate M⋆ with the entries
(M⋆ )ij = (Mij )⋆ , the transpose t M, by column-row interchange, with the entries (t M)ij = Mj i and
the adjoint M† = (t M)⋆ with the entries (M† )ij = (Mj i )⋆ . Given a matrix representation : G →
GL(d, C), by associating each element g of the group G with the complex conjugate ⋆ (g), the transpose
t
(g) and the adjoint † (g) of (g) one respectively defines the conjugate ⋆ , the transpose t and the
adjoint † of the matrix representation .
2.3 Direct sums
Let : G → GL(V, C) be a linear representation. A proper subspace V1 of the representation space V
by definition is stable or invariant under the group G iff
∀v1 ∈ V v1 ∈ V1 ⇒ (g)(v1 ) ∈ V1
∀g ∈ G
(2.13)
or, in terms of subsets, (g)V1 ⊆ V1 , ∀g ∈ G. A subspace V1 of V is proper iff it is distinct from V and
V and {0}
are trivially stable under any group G. The restriction
the zero-dimensional vector space {0}.
V1 (g) of (g) to V1 determines an automorphism of V1 and follows the group homomorphism rule
V1 (gh) = V1 (g) ◦ V1 (h) ∀(g, h) ∈ G, which means that the application
V1 : G → GL(V1 , C), g → V1 (g) s.t. V1 (g)(v1 ) = (g)(v1 ) ∀v1 ∈ V1
(2.14)
is a linear representation of the group G on the vector space V1 , which is called a subrepresentation
of .
−1 −1 =
−1 ) =
4 (g)(fˆ ) = fˆ (g)
ˆ s (g)sr Srm
= r ( s ( n fˆ n Sns ) ×
er ) Srm
= (g)( r eˆ r Srm
m
se
r
r (g)(ˆ
n n nm
−1 =
(g)sr ) Srm
n
fˆ n
s
r
−1 =
Sns (g)sr Srm
n
fˆ n (S (g)S −1 )nm .
00005-p.5
EPJ Web of Conferences
Select a basis {ˆem } in V1 and extend it to a basis {ˆem } ∪ {fˆ n } in V, which always is possible whenever
V is finite-dimensional or otherwise once the axiom of choice is allowed.5 A subspace V2 f of V is
linearly spanned by the set of vectors fˆ n . It is called a complement of the subspace V1 in the vector
space V, because any vector v in V writes uniquely as v = v1 + v2 with v1 ∈ V1 and v2 ∈ V2 f . It is
If V is finite-dimensional with dimension d and the dimension of V1
observed that V1 ∩ V2 f = {0}.
is d1 then the dimension of V2 f is d2 = d − d1 . If, conversely, a finite-dimensional vector space V
and
contains two subspaces, V1 with dimension d1 and V2 with dimension d2 , such that V1 ∩ V2 = {0}
d = d1 + d2 is the dimension of V then every v in V writes uniquely as v = v1 + v2 with v1 ∈ V1 and
v2 ∈ V2 . It may be emphasized that a complement of a proper subspace is a proper subspace and that
are the complements of each other in V. One symbolically formulate the fact that two proper
V and {0}
subspaces V1 and V2 of a vector space V are the complements of each other in V as
V = V1 ⊕ V2
(2.15)
In the event that not only the proper subspace V1 but also the selected complement V2 in V is stable
under the group G, the restriction V2 : G → GL(V2 , C) of to the representation space V2 makes up
another subrepresentation of . Importantly, ∀g ∈ G ∀v ∈ V, (g)(v ) is fully and uniquely determined
by the sum V1 (g)(v1 ) + V2 (g)(v2 ) with v1 ∈ V1 and v2 ∈ V2 . In addition, vi ∈ Vi ⇒ i (g)(vi ) ∈
Vi and vj ∈ Vj =i ⇒ Vi (g)(vj ) = 0 (i, j = 1, 2). It is customary to transcribe these properties by
symbolically equating to the direct sum of V1 and V2 :
= V1 ⊕ V2
(2.16)
With respect to the basis {ˆem } ∪ {ˆen }, built by union of the basis {ˆem } in V1 and the basis {ˆen } in V2 ,
the matrix representatives (g) of the linear operators (g) on V write in the block diagonal form
1
(g) 0
1
2
∀g ∈ G
(2.17)
(g) = (g) ⊕ (g) ≡
0 2 (g)
namely as the direct sum 1 (g) ⊕ 2 (g) of the matrix representatives 1 (g) of the linear operators V1 (g)
on V1 with respect to the basis {ˆem } and of the matrix representatives 2 (g) of the linear operators V2 (g)
on V2 with respect to the basis {ˆen }. Again, now to implicitly recall the block-diagonal structure of the
matrix representatives (g), it is the convention to symbolically write
= 1 ⊕ 2
(2.18)
and, subsequently, to state that the matrix representation is the direct sum of the sub-matrix
representations 1 and 2 .
As an illustration, let G : G → GL(V, C) be the regular representation of a group G on the vector
space V with basis {ˆeg }
g∈G and let V1 be the one-dimensional subspace of V consisting in the
scalar
ˆ
ˆh
e
.
V
evidently
is
stable
under
G:
∀
v
∈
V
∃!
a
∈
C
:
v
=
a
multiples of the vector g∈G
1
1
1
h∈G e
g 1
so that ∀g ∈ G G (v1 ) = a h∈G eˆ gh = v1 ∈ V1 . Let V2 be the subspace of V spanned by the nG − 1
vectors (ˆeh − eˆ e )h∈(G−{e}) , where e is the unit element of G and nG is the order of G. It easily is shown
that
V2 is nG − 1 and that
V1 ∩ V2 = {0}, by noticing that if a h∈G eˆ h =
the dimension of the subspace
eh − eˆ e ) then
eh − ( h∈G,h=e bh + a)ˆee = 0, ergo a = 0 and bh=e =
h∈G,h=e bh (ˆ
h∈G,h=e (bh − a)ˆ
0 ∀h ∈ G. Next, by applying the linear operators G (g) on the nG − 1 vectors (ˆeh − eˆ e )h∈(G−{e}) as
G (g)(ˆeh=e − eˆ e ) = eˆ gh − eˆ g = (ˆegh − eˆ e ) − (ˆeg − eˆ e ), it is straightforwardly inferred that the subspace
V2 is stable under G. Accordingly, the subspaces V1 and V2 thus constructed are effectively complement
5 The axiom of choice is not universally accepted because it leads to strange theorems, the most famous being the Banach-Tarski
paradoxical decomposition. Ignoring it however also leads to disasters, for instance a vector space may have no basis or may have
bases with different cardinalities. As to cure some of the inconveniences, in particular the existence of non-measurable sets of
reals, the axiom of determinacy was put forward in replacement, but this still might not be all satisfactory. Under this axiom every
subset of the set of reals R is Lebesgue-measurable, but, for instance, R as a vector space over the set of rationals Q has no basis.
00005-p.6
Contribution of Symmetries in Condensed Matter
of each other and invariant under G so that G can be put into the direct sum of the subrepresentations
built over these proper subspaces. Another choice of complement could have been made with the nG − 1
vectors eˆ h∈(G−{e}) , but this is not stable under G. It suffices to observe that G (g)(ˆeg−1 ) = eˆ e does not
belong to this complement.
2.4 Maschke’s theorem
A convenient tool to handle the direct sums of proper subspaces is the projection operator. It is
recalled that given the decomposition V = V1 ⊕ V2 f , every vector v in V by definition writes uniquely
as v = v1 + v2 with v1 ∈ V1 and v2 ∈ V2 f . The linear operator f that sends every vector v in V
onto its component v1 in V1 defines the projection operator of V onto V1 along V2 f . It is clear that
f ◦ f = f . It also is almost obvious that the image of f is im(f ) = {f (v ) | v ∈ V} = V1 , the
= V2 f and the restriction f V1 of f to V1 is the identity
kernel of f is ker(f ) = {v ∈ V | f (v ) = 0}
1V1 in V1 . Conversely, let f : V → V be a linear operator on V. If the dimension of im(f ) = V1 is
d1 and the dimension of ker(f ) = V2 f is d2 then d1 + d2 = d is the dimension of V. If in addition
f V1 = 1V1 , that is to say the restriction f V1 of f to V1 is the identity 1V1 in V1 , then V1 ∩ V2 f = {0}.
It then follows that V = V1 ⊕ V2 f . It again is clear that f ◦ f = f , which thus makes up another
equivalent definition of a projection operator f . A bijective correspondence is thus established between
the projection operators f of V onto V1 and the complements V2 f = ker(f ) of V1 in V.
Let : G → GL(V, C) be a linear representation of a finite group G on a finite-dimensional vector
space V over the field C of the complex numbers. Let V1 be a proper subspace of the representation
space V, which is invariant under the group G. Let V2 f be an arbitrary complement of V1 in V, not
necessarily invariant under the group G. Let f be the projection operator of V onto V1 bijectively
associated to V2 f . Let be the “average” of f over G, which is defined as:
=
1 (g) ◦ f ◦ (g −1 )
nG g∈G
(2.19)
where nG is the order of the group G. is a linear operator on V, since it is a function sum of functionally
composed linear operators on V. “commute” with G:
◦ (h) =
=
1 1 (g) ◦ f ◦ (g −1 h) =
(hg) ◦ f ◦ ((hg)−1 h)
nG g∈G
nG g∈G
1 (h) ◦ (g) ◦ f ◦ (g −1 h−1 h) = (h) ◦ nG g∈G
∀h ∈ G
(2.20)
by using the dummy transformation g → hg in the second equality and the identity (hg)−1 = g −1 h−1
in the third equality. It follows that V2 = ker() is a subspace stable under G: ∀h ∈ G, ∀v2 ∈
whence
V2 , ((h)(v2 )) = (h)((v2 )), but (v2 ) = 0 by definition of V2 , so that (h)((v2 )) = 0,
((h)(v2 )) = 0, that is to say (h)(v2 ) ∈ V2 . Next,
( ◦ f )V1 = 1V1
and
ker( ◦ f ) = ker(f ) = V2 f
so that
◦ f = f
(2.21)
Indeed, f V1 = 1V1 since f is a projection operator and v1 ∈ V1 ⇒ (g −1 )(v1 ) ∈ V1 by the invariance
of V1 under G, so that ∀v1 ∈ V1 , (g) ◦ f ◦ (g −1 ) ◦ f (v1 ) = (g) ◦ f ◦ (g −1 )(v1 ) = (g) ◦
(g −1 )(
u1 ) = v1 , ergo ∀v1 ∈ V1 , ( ◦ f )(v1 ) = v1 , that is to say ( ◦ f )V1 = 1V1 . f and are linear
= 0,
which means ker(f ) ⊆
operators and V2 f = ker(f ) so that ∀v2 ∈ V2 f ( ◦ f )(v2 ) = (0)
V1
ker( ◦ f ). If ( ◦ f )(
u1 ) = 0 and u
1 ∈ V1 then u
1 = 0 since ( ◦ f ) = 1V1 , which implies that
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ker( ◦ f ) ⊆ ker(f ) since V = V1 ⊕ V2 f . It finally is inferred that is a projection operator:
◦=◦
1 1 (g) ◦ f ◦ (g −1 ) =
(g) ◦ ◦ f ◦ (g −1 ) = nG g∈G
nG g∈G
(2.22)
by using the equation 2.20 in the second equality and the equation 2.21 in the third equality. Accordingly,
the G-invariant subspace V2 = ker() is a complement of the initially assumed G-invariant subspace V1 :
V = V1 ⊕ V2 .
A fundamental theorem is thus proven, the so-called Maschke’s Theorem, which states that
whatever the linear representation : G → GL(V, C) of a finite group G on a finite-dimensional
vector space V over the field C , to every invariant subspace V1 ⊆ V is associated an invariant
complement V2 ⊆ V
V. With the same proof arguments it is extended, for any finite group G, to any
finite-dimensional vector space V over any scalar field K of any characteristic char(K) that does not
divide the order nG of the group G, this merely by generalizing the average procedure in equation (2.19)
to K-summation and division by nG 1K , where 1K is the multiplicative unit of K. It is clear that if
nG ≡ 0 (mod char(K)) then this G-averaging cannot be defined since nG 1K = 0K , where 0K is the
additive unit of K.
2.5 Inner products
Another proof of Maschke’s Theorem can be forged using inner products, inspiring generalizations
to compact continuous groups G. An inner product on a vector space V over the field C designates
a two-arguments application ◦ | • : V × V → C, which is i- linear in the second argument (•):
u | a v + b w
= a u | v + b u | w
∀(a, b) ∈ C2 ∀(
u, v ) ∈ V2 , ii- conjugate symmetric: u | v =
⋆
2
It immediately follows from
v | u
∀(
u, v ) ∈ V and iii- positive definite: v | v > 0 ∀v ∈ V − {0}.
the two first properties (i and ii) that the inner product is antilinear in the first argument (◦):
a v + b w
|u
= a ⋆ v | u
|u
, ∀(a, b) ∈ C2 , ∀(
+ b⋆ w
u, v ) ∈ V2 . An inner product in other words
is a positive definite conjugate symmetric sesquilinear form.
A sesquilinear form is the generic name for any application : V × V → C which is antilinear
in the first argument and linear in the second argument. uniquely defines an antilinear application :
→ u
# ≡ (
u, •). Conversely, an antilinear application from a vector space V to its dual V#
V → V# , u
uniquely determines a sesquilinear form. is non degenerate iff is injective, which means ker(
) =
or (
The sesquilinear form defined as (
{0}
u, v ) = 0 ∀v ∈ V ⇔ u
= 0.
u, v ) = (v , u
)⋆ ∀(
u, v ) ∈ V2
is the conjugate symmetric to . If = ( = −) then is called an hermitian form (anti-hermitian
form).
A vector u
is orthogonal to a vector v with respect to a sesquilinear form iff (
u, v ) = 0. Let W
u ∈ V | ∀v ∈ W, (
u, v ) = 0} makes up a subspace of V, called
be a subspace of V. The set W⊥ = {
then the restriction W of to W is
the orthogonal to W with respect to in V. If W ∩ W⊥ = {0}
W
non degenerate, which means that the restriction of to W is injective. If, in addition, W is finitedimensional then W# is of the same dimension as W and W becomes a bijection. sends every v ∈ V
to a unique linear form v # ∈ V# , since it is an application. The restriction w
# of v # to W obviously is
#
∈ W, because W is a bijection. In
also unique. To the linear form w
finally corresponds a unique w
other words, to every v ∈ V is associated a unique w
∈ W such that (v , u
) = (w,
u
) ∀
u ∈ W, that is
to say v − w
∈ W⊥ . It follows that V = W ⊕ W⊥ . W⊥ then is called the orthocomplement of W in V.
in which case (v , v ) =
A sesquilinear form is positive definite iff (v , v ) > 0 ∀v ∈ V − {0},
⊥
0 ⇔ v = 0 whence W ∩ W = {0} whatever the finite-dimensional subspace W of V. Thus, to every
finite-dimensional subspace W of a vector space V over the field C endowed with an inner product is
associated an orthocomplement W⊥ in V.
Let be a linear operator on the vector space V. The transpose of is the linear operator t on the
dual space V# defined from the pointwise relation t (
u# )(v ) = u
# (v ). t (
u# ) is called the pullback of u
#
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along . If is invertible then t = (−1 )# . Let be a non degenerate sesquilinear form. A linear
operator † may be defined in V from the pointwise relation (
u, v ) = († u
, v ). It is called the
#
adjoint of with respect to . If the application : V → V , u
→ u
# ≡ (
u, •) is bijective, which
is the case only if the vector space V is finite-dimensional, then the adjoint of always exists, given as
† = −1 ◦ t ◦ .6 A sesquilinear form by definition is invariant with respect to a linear operator iff (
u, v ) = (
u, v ) ∀(
u, v ) ∈ V2 . Obviously this is the case iff is invertible and † = 1V , namely
†
−1
= . then is said unitary. The unitary operators are normal operators. A linear operator is
normal iff it commutes with its adjoint: ◦ † = † ◦ . It is diagonalizable and its eigen-spaces are
pairwise orthogonal (spectral theorem for the normal operators). Another subfamily of normal operators
are the self-adoint operators: † = .7
If V is finite-dimensional and {eˆi }i=1,...,d is the selected basis in V then ∀(
u, v ) ∈ V2 ,
d
d
u, v ) =
= i ui eˆ i and ∃! (v1 , . . . , vd ) ∈ C s.t. v = j vj eˆ j , so that (
∃! (u1 , . . . , ud ) ∈ C s.t. u
i,j u⋆i (ˆei , eˆ j ) vj = U† V, where U† (≡ t U⋆ ) is the complex conjugate row vector (u⋆1 , . . . , u⋆d ) and
V the column vector (v1 , . . . , vd ). The sesquilinear matrix with the entries ij = (ˆei , eˆ j ) uniquely
determines once the basis is given. is non degenerate iff Det() = 0. A basis {eˆi }i=1,...,d is
orthonormal with respect to iff = Id (d × d unit matrix). Let be a linear operator and denote
A the matrix representative of and A† the matrix representative of † in the {eˆi }i=1,...,d basis. The
⋆
⋆
pointwise relation (
u, v ) = († u
, v ) is transcribed into t U⋆ A V = t (A† U) V = t U⋆ t (A† ) V.
⋆
It follows that A = t (A† ) therefore t A⋆ t ⋆ = t ⋆ A† or else A† = (t ⋆ )−1 t A⋆ (t ⋆ ), since by
hypothesis is non-degenerate. If in addition the chosen basis is orthonormal with respect to then
A† = tA⋆ .
It is emphasized that inner products can be defined solely on vector spaces over the field R of the
real numbers, which is an ordered field, or the field C of the complex numbers, which is not ordered but
makes up an ordered extension of the field R. The basic reason is that otherwise it becomes meaningless
to require that a sesquilinear form be positive definite. This clearly excludes all the fields with non zero
characteristic, which cannot have an ordered subfield.
2.6 Unitarity and unitarisability
A linear representation : G → GL(V, C) of a finite group G on a vector space V over the field C by
definition is a unitary representation if the representation space V is endowed with an inner product
◦ | • : V × V → C which is invariant under G:
(g)(
u) | (g)(v ) = u | v ∀(
u, v ) ∈ V2 , ∀g ∈ G
(2.23)
which means that the linear operators (g) are unitary for every g in G. Another way telling the same
thing is that the linear representation commutes with the inner product ◦ | •.
6 It is customary in physics to use the so-called bra-ket notation. The space V then is endowed with an inner product ◦ | •
(pre-Hilbert space). V is complete for the associated norm (Hilbert space), namely every Cauchy sequence in V converges within
V. A vector is denoted by a ket | and a linear form by a bra |. The application of a linear operator O on a ket is described
as O|. Its dual is applied on a bra | as |O# to mean (|O# )| = |(O|). Its adjoint is applied on a ket | as O† |
so that |(O† |) = (|O# )|⋆ . To any ket | one may associate a bra | (Riez Theorem). The converse is true solely in
finite dimension. If V is infinite-dimensional then V can be put in bijection only with the subspace of continuous linear forms in
the dual V# . The “discontinuous” bra have no ket counterpart.
7 A bijective correspondence exists between the self-adjoint operators H on a Hilbert space V and the families of unitary operators
U()∈R on V with the group property U( + ) = U() ◦ U() and the continuity property U( → ) → U(), to be precise
U() = exp(iH) (Stone’s theorem). When the Hilbert space is separable it suffices to assume weak measurability instead of
continuity (von Newman). This bijection is useful in establishing the uniqueness of the irreducible unitary representation of the
algebra of canonical commutation relations on finitely many generators (Stone-von Newman theorem). This is no more the case
with infinitely many generators, concretely in quantum field theory where in general there is no unitary equivalence between
canonical commutation relation representation of the free field and that of the interacting fields (Haag theorem).
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If the representation space V is finite-dimensional with dimension d then a unitary matrix
representation ϒ : G → GL(d, C) of the group G is obtained by selecting in V an orthonormal basis
{eˆi }i=1,...,d with respect to the inner product ◦ | •. ϒ associates each element g of the group G to a
unitary matrix representative ϒ(g):
ϒ(g)† ϒ(g) = (t ϒ(g)⋆ ) ϒ(g) = Id
∀g ∈ G
(2.24)
where Id is the d × d unit matrix, or else ϒ(g)† = ϒ(g −1 ) for every element g in the group G.
Let W be a finite-dimensional proper subspace of the representation space V and let W⊥ be
the orhocomplement of W in V. One has v ∈ W⊥ ⇔ v | w
= 0 ∀w
∈ W and V = W ⊕ W⊥ , by
⊥
definition of W . Assume that W is invariant under G. It follows, by the equation (2.23) that
= 0 ∀w
∈ W for every g ∈ G or else, choosing u
= (g −1 )(w)
and
v ∈ W⊥ ⇔ (g)(v ) | (g)(w)
⊥
making use of the G-invariance of W, v ∈ W ⇔ (g)(v ) | u
= 0 ∀
u ∈ W, which merely means that
W⊥ is invariant under G. A sub-representation of a unitary representation is obviously unitary for
the restricted inner product. Accordingly, every unitary representation of a finite group G on a vector
space V over the field C that contains a finite-dimensional subspace W invariant under G can be
decomposed into two unitary subrepresentations as
= W ⊕ W⊥
(2.25)
where W stands for the restriction of to W and W⊥ for the restriction of to the orthocomplement
W⊥ of W in V. The two subrepresentations might in turn be decomposed into subrepresentation and
so on. The process must end after a finite number of iterations if V is finite-dimensional, since by
hypothesis the invariant subspace is a proper subspace so that at each step the dimension of the
subrepresentation spaces to consider is decreased. It nevertheless is emphasized that no conditions is
imposed on the dimension of the representation V, which thus might be infinite. So, at least as far
as G is finite, the dichotomy processes might go on indefinitely and lead to infinite direct sums or even
direct integrals. As a matter of fact, the construction of a meaningful direct integral often can fail, all the
more as the group G is unspecified, and leads to extremely delicate and difficult problems of functional
analysis.
A linear representation : G → GL(V, C) is unitarisable by definition if an inner product invariant
under G can be
defined in the representation space V. Assume that V possesses a basis {ˆei }. Whatever
the
vector
u
=
u)ˆei in V the set of complex numbers {xi (
u)} is uniquely defined. So is the product
i xi (
⋆
,
(
u
)x
(
v
)
=
u
|
v
which
thus
determines
an
application
V × V → C, conjugate symmetric,
x
i
i i
u) |2 > 0 unless xi (
u) = 0 ∀i). In other
linear in the second argument and positive definite ( i | xi (
words, an inner product ◦ | • in V is defined by declaring that the basis {eˆi } is orthonormal.8 If the
group G is finite then the application
◦ | •G : V × V → C, (
u, v ) → u | v G =
g∈G
(g)(
u) | (g)(v )
(2.26)
can always be defined. It is straightforwardly shown that i- ◦ | •G is linear in the second argument
because ◦ | • is linear in the second argument and (g) is a linear operator on V for every g ∈ G,
ii- ◦ | •G inherits from ◦ | • the conjugate symmetry property, and iii- ◦ | •G is positive definite
8 Every regular representation of a finite group G for instance is obviously unitarisable. It is made unitary merely by declaring
G
that the G-indexed basis vectors eˆ h are orthonormal in the representation space.
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Contribution of Symmetries in Condensed Matter
because the sum of strictly positive numbers is strictly positive. It further is found out that
(g)(
(h) {(g)(
u) | (g)(v )G =
u)} | (h) {(g)(v )}
h∈G
=
h∈G
(hg)(
u) | (hg)(v ) =
= u | v G
k∈G
(k)(
u) | (k)(v )
∀g ∈ G, ∀(
u, v ) ∈ V2
(2.27)
In other words, ◦ | •G is an inner product which is invariant under G. The linear representation
becomes a unitary representation by endowing the representation space V with the inner product
◦ | •G . Note that every change of inner products is equivalent to a basis change.9
A fundamental theorem is thus proven, which states that every linear representation of a finite
group G on a vector space V over the field C is unitarisable and therefore isomorphic to a unitary
representation. It thus can always be decomposed into subrepresentations whenever there exists a
finite-dimensional proper subspace invariant under G in the representation space. The group average
displayed in the equation (2.26) is the so-called Weyl’s Trick. It already was employed in a disguised
manner for a projection operator in the equation (2.19). It can be extended to linear representations
of topological groups,10 provided the summation over the group elements can be generalized to an
appropriate integration.11
One finally may wonder whether the unitarity concept is worth extending to invariance with respect
to hermitian forms not necessarily positive definite, to deal with linear representations on vector spaces
9 A basis {fˆ } orthonormal with respect to ◦ | • can even always be built, using for instance the Gram-Schmidt procedure:
i
G
si
fˆ i = si | si G
with
sˆ1 = eˆ 1
and sn = eˆ i −
n eˆ n | sˆj G
sˆj
sˆ | sˆj G
j =1 j
(n > 1)
(2.28)
Of course, the change from the basis {ˆei } to the basis {fˆ i } describes nothing but a similarity transformation.
10 A topological group by definition is a set G endowed with a group structure and a topological structure such that the group
operation Gop : (g, h) → gh−1 is a continuous function, to be precise the inverse image of any open set of G by this function
is an open set of the topological product space G × G. A topological space is separated iff for any pair of distinct points
there exists disjoint neighborhoods (Hausdorff). It is quasi-compact iff a finite cover can be extracted from every open cover
(Borel-Lebesgue). It is compact iff it is separated and quasi-compact. It is locally compact iff every point possesses a compact
neighborhood. It is simply connected iff every loop is homotopic to the null loop. A loop is a continuous function : [0, 1] → G
such that (0) = (1). A loop at a point g is null iff im() = {g}. A loop is homotopic to a loop iff there exists a continuous
function : [0, 1] × [0, 1] → G such that (0, ) = (1, ) ∀ and (, 0) = (), (, 1) = () ∀. A topological group is
m-connected iff at every point it shows m homotopy classes of loops. Its representations then might be m-valued, but for each
multiply-connected group there exists a simply connected group, the universal cover, that is homomorphic to it. A few examples:
SU (n) is compact simply connected. SO(n) is compact 2-connected and its universal cover is Spin(n). Spin(3) is isomorphic to
SU (2). O(p, q) (0 < p ≤ q) is non-compact 4-connected. . . .
A field is topological iff its additive and multiplicative groups are topological. A vector space on a topological field
endowed with a topological structure such that the vector addition and the scalar multiplication are continuous is topological.
A continuous representation of a topological group G on a topological vector space V over the field C is a linear representation
: G → GL(V, C) such that the function r : G × V → V, (g, v ) → r(g, v ) = (g)(v ) is continuous on the two variables g ∈ G
and v ∈ V.
11 If G is a locally compact topological group then there always exist a measure dg and only one carried by G and enjoying
the properties i- G F(g)dg
= G F(gh)dg for every h in G and every continuous function F on G (invariance of dg under
and ii- G dg = 1 (mass normalization). If G is compact then dg is also invariant under left translation:
right translation)
G F(g)dg = G F(hg)dg, in which case dg is called the bi-invariant or Haar measure of G. If the group G is finite of order
nG , the measure dg is obtained by assigning to each g in G a mass equal to 1/nG . If G is the group SO(2) of the planar
rotations and if every g ∈ SO(2) is represented in the form g ≡ exp(i) ( taken modulo 2) the invariant measure is d/2.
As a matter of fact, the concrete construction of the Haar measure generally is far from being obvious, except possibly for
groups of geometric nature (O(n, K), SO(n, K), U (n, K), . . .). An efficient method can be worked out for a Lie group G of
dimension n represented by unitary matrices U= exp(iH) of order N . The hermitian matrix H belongs to the associated Lie
algebra G and can be parametrized as H(x) = p xp Xp with xq = Tr(HXq ), by means of the generators Xp chosen such that
Xp , Xq = iCpqr Xr and Tr(Xp Xq ) = pq . As from the invariant metric Tr(dU † dU) = −Tr U −1 dUU −1 dU = pq (x)dxp dxq
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over fields with non zero characteristic. A more generalized approach might even be considered, since
sesquilinear forms might be defined on any module over a ring for an unspecified antiautomorphism
(in place of the conjugate complex involution). The drawback is that the crucial result according to
which every proper subspace possesses an orthocomplement then would be lost. Isotropic subspaces,
the vectors of which are all orthogonal to at least one of their own non null vectors, might exist, that
thus might not necessarily have a complement.
2.7 Irreducibility and reduction
A linear representation of a group is said irreducible if its representation space contains no proper
invariant subspace under the action of the group and reducible otherwise. A reducible representation is
not necessarily decomposable into subrepresentations, since this requires that to the identified invariant
subspace is associated an invariant complement. A linear representation then might be reducible but
indecomposable. A linear representation is said completely reducible if it is decomposable down to
irreducible components.
Let : G → GL(V, K) and : G → GL(W, K) be two linear representations intertwined with the
isomorphism : V → W. Assume that there exists a G-invariant subspace V1 in V and denote W1 its
1 ∈ W1 ⇒ −1 (w
1 ) ∈ V1 ⇒
image by in W. W1 obviously is a subspace of W, which is G-invariant: w
−1
−1
1 ) ∈ V1 ⇒ ( ◦ ◦ )(w
1 ) = (g)(w
1 ) ∈ W1 . It follows that every linear representation
( ◦ )(w
isomorphic to a reducible linear representation is itself reducible. If V2 is a G-invariant complement to
V1 in V then its image W2 by is a complement of W1 in W. Indeed, the restriction of to Vi (i = 1, 2)
defines two isomorphisms i : Vi → Wi (i = 1, 2) so that v ∈ V1 ∩ V2 ⇔ (v ) ∈ W1 ∩ W2 and the
dimensions of Vi and Wi (i = 1, 2) are the same. W2 of course is also G-invariant. This means that
every linear representation isomorphic to a decomposable linear representation is itself decomposable.
Assume now that there is no G-invariant subspace V1 in V then obviously there can be no invariant
subspace in W, otherwise its image by −1 would be a G-invariant subspace in V in contradiction
with the hypothesis. Accordingly, every linear representation isomorphic to an irreducible linear
representation is itself irreducible. It similarly is shown that every linear representation isomorphic to
a reducible but indecomposable linear representation is itself reducible but indecomposable and every
and the identity d(eK ) =
G
1
0
dze(1−z)K dKezK it is inferred that
F(U) dU =
Det((x)) F(x) dx
pq (x) =
with
X
1
−1
(1 − |z|) exp(z
xr Crqp ) dz
r
(x) is diagonalized
by the same unitary matrix as the n × n real antisymmetric matrix M(x) = −M† (x) with the entries
Mpq (x) = r xr Crqp . It follows that if ±ij (j ∈ R+ ) denotes the eigenvalues of the matrix M(x) then
Det((x)) =
sin2 (i /2)
(i /2)2
i
The eigenvalue problem M(x)v = iv is equivalent to solving the equation [V, H(x)] = V with V = p vp Xp . It is observed
that S † (V)S = S † (VH(x) − H(x)V)S = S † (VSS † H(x) − H(x)SS † V)S. Thus, if S is the matrix that diagonalizes H(x) then
(S † VS)ij (νi − νj − ) = 0: the eigenvalues k of M(x) needed to evaluate the Haar measure are differences of eigenvalues νi of
H(x).
Assume that : G → GL(V, C) is a linear representation of a compact group G and assume that the representation space is
endowed with an inner product ◦ | •. The quantity u | v G = G (g)(
u) | (g)(v ) dg (Weyl’s Trick) is well defined since
G is compact and g → (g)(
u) | (g)(v ) is continuous.
It is clearly Hermitian and it is G-invariant since the Haar measure is
right invariant. It finally is positive definite: v | v G = G (g)(v ) | (g)(v ) dg > 0, ∀v = 0 since (g)(v ) | (g)(v ) > 0. We
thus have demonstrated that every linear representation of a compact group is unitary. Using similar arguments as with the
unitary representation of finite groups it then is shown that every finite-dimensional linear representation of a compact group is
completely reducible. As a matter of fact, as far as only the finite-dimensional representations on the vector spaces over the field
C are considered, almost all the theorems that are proved for finite groups are safely extended to compact groups, be it that at
some places a sum must be replaced by an integral.
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linear representation isomorphic to a completely reducible linear representation is itself completely
reducible. It is the usage also to call irreducible (resp. reducible and decomposable, reducible but
indecomposable, completely reducible) the matrix representation obtained from an irreducible (resp.
reducible and decomposable, reducible but indecomposable, completely reducible) linear representation
by selecting a basis in the representation space.
Complete Reducibility Theorems may be formulated for certain families of linear representations.
Among the most important for the physics of the finite groups of symmetry is the one which states that
every linear representation of a finite group on a finite-dimensional vector space over the field of
complex numbers is completely reducible. As to prove it one proceeds by induction on the dimension d
of the representation space V. Assume that the statement holds for all the representations of dimension
smaller than d, and let be a linear representation of dimension d. If V is irreducible, then there
is nothing to prove. Otherwise, there exists a proper subspace V1 , therefore of dimension d1 < d,
invariant under G. According to the Maschke’s Theorem, V1 has in V a complement V2 , therefore
of dimension d2 < d, which is also invariant under G. Accordingly, = 1 ⊕ 2 , where i (i = 1, 2) is
the restriction of to Vi (i = 1, 2). Now, by the induction hypothesis the subrepresentation i (i = 1, 2)
is completely reducible, since di < d (i = 1, 2). So the same is true of , which ends the proof. Note that
although the mathematical induction might suggest that the theorem might be true for infinite countable
dimension, the corresponding extension would make up an abuse at this step for the Maschke’s Theorem
is demonstrated only for finite-dimensional V.
The theorem is straightforwardly extended to the linear representation of the finite groups on the
finite-dimensional vector spaces over the fields whose characteristic does not divide the order of the
group, from the corresponding extension of the Maschke’s Theorem. Using the Weyl’s Trick the theorem
also is extended to the linear representation of the compact groups on the finite-dimensional vector
spaces over the field C. Note, meanwhile, that the finite groups are compact, for the discrete topology. It
happens that finally the infinite-dimensional case does not cause excessively more troubles for compact
groups. It indeed is shown that every continuous representation of a compact group on a Hilbert
space V, be it infinite-dimensional, is isomorphic to the Hilbert sum of finite-dimensional unitary
representations and the set of G-finite vectors is dense in V. A Hilbert sum of unitary
representations
ˆ : G → ⊕
ˆ V = {(v ) | v ∈ V ∧ v 2 < ∞} on
: G → V is the unitary representation ⊕
ˆ V is the
the Hilbert sum of the representation spaces V , that coincides with on each sector. ⊕
u , v and contains ⊕ V as a dense subspace
Hilbert space with inner product ((
u ), (v )) = with V ⊥V∀= . A set of G-finite vectors is the set of all vectors vf in in V such that the dimension of
the vector space spanned by {(g)(vf in ), g ∈ G} is finite. It follows in particular that the irreducible
unitary representations of the compact groups are all finite dimensional. A proof is provided first by
showing that there always exists a finite-dimensional G-invariant (closed) subspace in V, for instance the
eigenspace of any non zero eigenvalue of a G-averaged compact operator on V, and next, using the Zorn’s
ˆ V , partially ordered by inclusion, necessarily shows a maximal
Lemma, by establishing that the set ⊕
ˆ
ˆ V )⊥ in
element. As a result ⊕ V cannot be different from V, otherwise there would exist V ∈ (⊕
violation of the maximality. Note that the “Zorn’s Lemma” is equivalent to the axiom of choice (see
footnote 5). Non compact groups do show infinite-dimensional representations which are more delicate
to handle or else linear representations that cannot be isomorphic to unitary representations or reducible
representations that are indecomposable.12
12 Although to some extent either exotic or pathological for what might concern physical systems the counterexamples to the
complete reducibility of the linear representations are not that uncommon, even with finite groups, and it always is instructive to
have scrutinized at least one. Consider for instance the matrix representation
1k
: Cp = s | s p = e → GL(2, Z/qZ), s k →
01
of the cyclic group Cp of order p and generator s on the linear group of the 2 × 2 invertible matrices with entries in the field Z/qZ
of characteristic char(Z/qZ) = q. At first it is observed that if q does not divide p then cannot be a group homomorphism and
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Now, let : G → GL(V, C) be a completely reducible linear representation of a finite group G.
Choose an initial G-invariant subspace, find its complement and perform a first decomposition into two
sub-representations, then proceed similarly on each of these and so on until getting only irreducible
sub-representations. Grouping
isomorphic irreducible summands, one most generally would write
= 1 ⊕ 2 ⊕ . . . ⊕ s = k k , where k is isomorphic to the direct sum of nk copies of an irreducible
linear representation k : G → GL(Vk , C), these by construction being non-isomorphic for different k’s.
A symbolic manner transcribing all this is
V∼
=
k
k
V⊕n
k
and
∼ n1 1 ⊕ n2 2 ⊕ · · · ⊕ ns s ≡
nk k
(2.29)
k
k
where V⊕n
is isomorphic to the subspace Xk of V spanned by the different G-invariant subspaces
k
defines the multiplicity of the irreducible component
of V associated with each copy of k and nk k contained in . It is customary to call = k k the canonical decomposition of , or else the
decomposition of into isotypical components k . An irreducible matrix representation k : G →
GL(dk , C) is associated with the irreducible linear representation k : G → GL(Vk , C) as soon as a
basis is selected in the representation space Vk . With every isomorphism of V that transforms a
given copy of Vk in V to another copy of Vk in V is associated two distinct bases in one-to-one
⊕n
correspondence and two isomorphic irreducible matrix representations. A basis of Xk ∼
= Vk k thus
may be built from different isomorphisms in V sending an initial copy of Vk in V to the different
copies of Vk in V. With respect to this basis the linear representation k is associated to a matrix
representation k : G → GL(nk dk , C) isomorphic to the direct sum of nk copies of the irreducible matrix
representations k : G → GL(dk , C). A basis in V is obtained from the union of the bases built on each
⊕n
subspace Xk , since V is the direct sum of the Xk ∼
= Vk k . The matrix representation : G → GL(d, C)
associated with the linear representation :
G → GL(V, C) with respect to this basis in V is given as
the direct sum = 1 ⊕ 2 ⊕ . . . ⊕ s = k k . It again is standard to write
∼ n1 1 ⊕ n2 2 ⊕ · · · ns s =
nk k
(2.30)
k
and customary to call = k k the canonical decomposition of , or else the decomposition of into isotypical components k . A similar procedure may be replicated to get canonical decompositions
of linear representations of compact groups, possibly by using Hilbert sums of representations. Note
that at this stage it is not sure whether the canonical decomposition is unique, so deserves its name, and
whether the nk are unambiguously defined.
therefore cannot be a matrix representation associated with a linear representation. Next ker() = {e}, that is to say is injective,
iff q = p. Now assuming that either q divides p or equals p, the one-dimensional space spanned by the (1, 0) vector is invariant
under Cp , but it has no invariant complement: the representation is reducible but indecomposable. In a different context, if l
is a prime then the set Zl = inv.lim.Z/ l n Z of l-adic integers makes up a compact topological group, which has the continuous
reducible but indecomposable representation
: Zl → GL(2, Ql ), x →
1x
0 1
on a 2-dimensional vector space over the field Ql of l-adic numbers. This example tells that “compact group” and “continuous
representation” are not enough conditions. The basis field must be C. Substituting the additive group R for Zl and the
automorphism group GL(2, C) for GL(2, Ql ) a third example of continuous representation is obtained, which again is reducible
but indecomposable. It also is not unitarizable. In this case the failure of complete reducibility is to ascribe to the fact that R is
not compact. It is only locally compact, because it is not bounded. The compact subsets of Rn (Cn ) are the closed and bounded
subsets of Rn (Cn ).
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2.8 Schur’s lemmas
It is clear that there exists a number of ways to decompose reducible linear representations down
to irreducible components, so that to proceed further it is necessary to get deeper insights into their
isomorphisms. As a matter of fact, the irreducible linear (or matrix) representations are special in their
intertwining. This is formulated in the Schur’s Lemmas:
Let 1 : G → GL(V1 , C) and 2 : G → GL(V2 , C) be two irreducible representations of a finite
group G and let HomG (V1 , V2 ) = { : V1 → V2 | 2 (g) ◦ = ◦ 1 (g) ∀g ∈ G} be the vector space of
intertwining operators from V1 to V2 . Then, denoting dHomG (V1 ,V2 ) the dimension of HomG (V1 , V2 ),
Schur 1 − 1 ∼ 2 ⇐⇒ dHomG (V1 ,V2 ) = 1
Schur2 − 1 ≁ 2 ⇐⇒ dHomG (V1 ,V2 ) = 0
A proof is provided by observing that ker() = {v ∈ V1 | (v ) = 02 } is a G-invariant subspace of
V1 and im() = {(v ) | v ∈ V1 } is a G-invariant subspace of V2 : v ∈ ker() ⇒ (1 (v )) = 2 ((v )) =
u ∈ V1 : (
u) = v ⇒ ∃w
= 1 (
u) ∈ V1 : (w)
=
2 (02 ) = 02 ⇒ 1 (v ) ∈ ker() and v ∈ im() ⇒ ∃
u)) = 2 ((
u)) = 2 (v ) ⇒ 2 (v ) ∈ im(). The irreducibility of 1 and 2 leaves ker() = {01 }
(1 (
or V1 and im() = V2 or {02 } as the only options. is non zero iff ker() = {01 }, which means
that is injective, and im() = V2 , which means that is surjective, that is to say iff is an
isomorphism. As a consequence, 1 ∼ 2 ⇔ V1 ∼
= V2 ⇔ HomG (V1 , V2 ) = {0}, which partially proves
Schur 1, and 1 ≁ 2 ⇔ V1 ≇ V2 ⇔ HomG (V1 , V2 ) = {0}, whence dHomG (V1 ,V2 ) = 0, which ends the
proof of Schur 2.
V1 ∼
= EndG (V2 ), where EndG (Vi ) (i = 1, 2) is the vector
= EndG (V1 ) ∼
= V2 ⇔ HomG (V1 , V2 ) ∼
space of the endomorphisms i (i = 1, 2) of Vi (i = 1, 2) that commute with G : i (g) ◦ i = i ◦
i (g) (i = 1, 2) ∀g ∈ G. Unlike HomG (V1 , V2 ≇ V1 ), which is only a vector space, EndG (Vi ) (i = 1, 2),
endowed with the canonical composition law ◦ for the functions, shows the structure of a division
algebra, with unit ǫi (i = 1, 2) and composition inverse for each of its non zero elements. Now, select a
non zero 1 in EndG (V1 ) and pick up another arbitrary ∈ EndG (V1 ). Obviously ◦ 1−1 ∈ EndG (V1 ). It
is implicitly assumed that the representation space V1 is finite-dimensional. Accordingly, as the field C
is algebraically closed, there always exists for ◦ 1−1 an eigenvalue ∈ C: ker( ◦ 1−1 − ǫ1 ) = {0}.
−1
−1
∀g ∈ G and ∀v ∈
On the other hand, [( ◦ 1 − ǫ1 ) ◦ 1 (g)](v ) = [2 (g) ◦ ( ◦ 1 − ǫ1 )](v ) = 0,
ker( ◦ 1−1 − ǫ1 ): ker( ◦ 1−1 − ǫ1 ) is a G-invariant subspace of V1 . The irreducibility of 1 then
implies that ◦ 1−1 − ǫ1 = 0, that is to say = 1 or else EndG (V1 ) ≡ C1 . In other words 1 ∼ 2
iff every intertwining operator from V1 to V2 is isomorphic to an endomorphism of V1 proportional to
1 , whence dHomG (V1 ,V2 ) = 1, which ends the proof of Schur 1.
Schur’s Lemma are straightforwardly generalized to finite-dimensional irreducible representations
of compact groups, using the same proof arguments. With infinite-dimensional representations discrete
eigenvalues might not necessarily exist and one has to resort to the spectral theorem for normal bounded
operators,
which states that for any in EndG (V) there exists a projection valued measure such
that = spec() d and that the only bounded endomorphisms of V commuting with are the ones
commuting with the self-adjoint projection (B) for each Borel subset B of the spectrum spec().
Whatever the case, Schur 1 obviously implies that
: G → GL(V, C) is irreducible ⇐⇒ HomG (V, V) ≡ EndG (V) ∼
= C 1V
Schur’s Lemma may be extended to scalar fields K other than the field C of complex numbers under
the weaker formulation:
1 ∼ 2 ⇐⇒ dHomG (V1 ,V2 ) = dEndG (Vi ,K) (i=1,2) = 0
1 ≁ 2 ⇐⇒ dHomG (V1 ,V2 ) = 0
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which is inferred solely from the G-invariance of the subspaces ker() and im() for any in
HomG (V1 , V2 ) and the irreducibility of 1 and 2 . It also is clear that any non zero in HomG (V1 , V2 )
is an isomorphism and therefore HomG (V1 , V2 ∼
= EndG (Vi , K) (i = 1, 2), endowed with the
= V1 ) ∼
canonical composition law ◦ for the functions, shows the structure of a division algebra over the
field K. This leads to three possibilities: i- if K is algebraically closed then dEndG (Vi ,K) (i=1,2) = 1 and
EndG (Vi , K) (i = 1, 2) ∼
= C 1V , ii- if K is real closed, that is to say if K is not algebraically
= K 1V ∼
closed but its closure is a finite extension, then by virtue of the (1,2,4,8)-Theorem on the real division
algebras and since it implicitly is clear that EndG (Vi , K) (i = 1, 2) is associative but not necessarily
commutative, dEndG (Vi ,K) (i=1,2) may take the values 1, 2, 4 and the division algebra EndG (Vi , K) (i = 1, 2)
may be isomorphic to either R 1V , C 1V or Q 1V , where Q stands for the field of quaternions.13 iii- if K
is neither algebraically closed nor real closed then dEndG (Vi ,K) (i=1,2) is the square of an integer.
The transcription of the Schurs Lemmas into the language of complex matrix representations of
finite groups is easily inferred as:
Schur 1 - If : G → GL(d, C) is an irreducible complex matrix representation of dimension d of a
finite group G then every d × d matrix A commuting with is a multiple of the d × d identity
matrix 1d :
{ irreducible} ∧ {(g) A = A (g) ∀g ∈ G} ⇒ ∃ ∈ C : A = 1d
Schur 2 - No intertwining may exist between two irreducible complex matrix representation of a finite
group G except if these are associated with isomorphic representation spaces:
{1,2 irreducible} ∧ {2 (g) A = A 1 (g) ∀g ∈ G} ⇒ {A = 0 or 1 ∼ 2 }
Schur’s Lemmas have a number of impacting outcomes. Schur 1 for instance implies that every
irreducible complex representation : G → GL(V, C) of an abelian group G is 1-dimensional: ∀g ∈
G, since G is abelian, (g)(h) = (gh) = (hg) = (h)(g) ∀h ∈ G, whence, since is irreducible,
∃g ∈ C : (g) = g 1V , by Schur 1. It follows that ∀v ∈ V (g)(v ) = g v ⇒ (g)(v ) ∈ Span(v ), that
is to say every 1-dimensional subspace Span(v ) = {a v | a ∈ C} of V necessarily is G-invariant.
The irreducibility of then implies that the representation space V itself is 1-dimensional. This
easily is generalized to compact groups using similar arguments,14 but fails with scalar fields K
that are not algebraically closed. A simple illustration is provided by the real representations :
C3 = s| s 3 = e → GL(V, R) of the cyclic group C3 . If is irreducible then it either is isomorphic
to the 1-dimensional trivial representation or to the 2-dimensional representation that associates the
generator s of C3 to the 2-dimensional geometric rotation by an angle 2/3 in a plane. The matrix
representative of this rotation with respect to any selected basis in V has complex eigenvalues. It thus
13 The (1,2,4,8)-Theorem can be given different equivalent formulations. It in particular states that, up to isomorphism, the
only division algebra over a real closed field are the 1-dimensional real algebra R, the 2-dimensional complex algebra C, the
4-dimensional quaternion algebra Q and the 8-dimensional octonion algebra O. At each increase of the algebra dimension an
essential property is lost: a nonidentical involution must be introduced to get C, commutativity is lost with Q then associativity
is lost with O, but these algebra still are alternative. Algebras of higher dimension are constructed using the dimension-doubling
Cayley-Dickson process: (x1 , x2 )(y1 , y2 ) = (x1 y1 − y2 x2⋆ , x1⋆ y2 + y1 x2 ), (x1 , x2 )⋆ = (x1⋆ , −x2 ). According to this, the next in the
list is the 16-dimensional sedenion algebra S, which is no more alternative nor a division algebra, but retains the property of
power associativity. The (1,2,4,8)-Theorem encompasses the weaker previous Frobenius’, Hurwitz’s and Zorn’s Theorems on the
real division algebras, but unlike these is not proved algebraically. It actually emerges as a corollary to a theorem of topological
nature: the existence of an arbitrary division algebra of dimension n over the reals implies parallelizability of the sphere Sn−1
but according to the Bott-Milnor-Kervaire Theorem spheres are parallelizable only in dimensions n = 1, 2, 4, 8 (a manifold is
parallelizable iff the tangent space at each point stay isomorphic to its transform induced by any parallel transport along a curve).
There exists a variety of other avatars of the (1,2,4,8)-Theorem, in Topology (Hopf bundles over the spheres Sn , . . .), in Geometry
(construction of exceptional Lie algebra, . . .), in Number Theory (a sum of n squares of integers times another sum of n squares
of integers is a sum of n squares of integers iff n = 1, 2, 4, 8, . . .), . . . .
14 A number of way exists to establish that all the irreducible representations of a compact group are 1-dimensional iff G is
abelian. One may use for instance the fact that the commutator group CG = {ghg −1 h−1 | g, h ∈ G} = {e} iff G is abelian and that
this acts trivially on 1-dimensional representations.
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Contribution of Symmetries in Condensed Matter
cannot be diagonalized with only real entries in the diagonals. As a matter of fact, it can be shown that
the irreducible representations : G → GL(V, K) of an abelian group G are 1-dimensional over the
endomorphism ring EndG (V, K), which makes up an extension field of the field K.
Schur 2 allows demonstrating that
the canonical decomposition
of completely reducible linear
representations is unique. Let = k k and = k k be canonical decompositions of two
linear representations : G → GL(V, C) and : G → GL(U, C). Any in HomG (U, V) maps the
⊕n
⊕m
representation space Zk ∼
= Vk k of k , because every
= Uk k of k to the representation space Xk ∼
restriction kq of from a copy of Uk to a copy of Vq intertwines with two irreducible representations so
is nullas soon as k = qby virtue of Schur 2. In the more intuitive language of matrix representations, if
= k k and = k ϒ k are two canonical decompositions and if and are intertwined with
a matrix S then this cannot contain a non null off-diagonal block Sk,q=k with which the isotypical
components k of and ϒ q=k of would be intertwined. It follows, by taking for an irreducible
representation k : G → GL(Vk , C), that every sub-representation of which is isomorphic to an
irreducible representation k is contained in k , which gives an intrinsic description of k as isomorphic
to the direct sum of all the copies of k contained in . Accordingly, the canonical decomposition does
not depend on the manner it might be performed, which proves its uniqueness.
Another consequence of the Schur’s Lemmas, of utmost practical relevance for irreducible matrix
representations, is the so-called Orthogonality Theorem. Whatever the two irreducible representations
k : G → GL(Vk , C) and q : G → GL(Vq , C) of a finite group G and the linear application from Vq
to Vk , the average of over the group G, which is defined as
1 k (g) ◦ ◦ q (g −1 )
(2.31)
=
nG g∈G
is an intertwining operator: k (h) ◦ = ◦ q (h) ∀h ∈ G.15 In other words, ∈ HomG (Vq , Vk ). It
then follows from the Schur’s Lemmas that k ∼ q ⇔ = 1Vk ∼
=Vq and k ≁ q ⇔ = 0. =
Tr [] /Tr 1Vk , since
Tr [] =
k ∼q 1 1 Tr k (g) ◦ ◦ q (g −1 ) =
Tr q (g) ◦ ◦ (q (g))−1 = Tr []
nG g∈G
nG g∈G
and Tr 1Vk = dk , where dk is the dimension of k . Now, selecting a basis in Vk and a basis in Vq , the
linear representations k and q and the linear operators and get associated respectively with matrix
representations k and q and dk × dq (k lines − q columns) matrices T and S. In terms of matrix
elements of the corresponding matrices the equation (2.31) writes:
1 k
q
(g −1 )
(2.32)
(g) Tlm mn
Sj n =
nG g∈G lm j l
which comes out as a linear form with respect to the variables Tlm . If k ≁ q , that is to say if k = q,
then
for all systems of values of the Tlm . Its coefficients therefore are null, whence
thisk form vanishes
q
−1
(g)
(g
)
=
0 for arbitrary j , l, m, n. If k ∼ q , that is to say if k = q, then Sj n = j n
mn
g∈G j l
15
k (h) ◦ ◦ (q (h))−1 =
=
1 k (h) ◦ k (g) ◦ ◦ (q (g))−1 ◦ (q (h))−1
nG
g∈G
1 k (hg) ◦ ◦ (q (hg))−1 = nG
g∈G
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with = Tr [] /dk = (1/dk )
lm
lm Tlm , whence
1 k
q
(g −1 ) =
(g) Tlm mn
nG g∈G lm j l
1 lm Tlm j n
dk lm
(2.33)
q
which, by equating the coefficients of the Tlm , gives n1G g∈G jk l (g) mn (g −1 ) = d1k if l = m and j = n
q
and n1G g∈G jk l (g) mn (g −1 ) = 0 otherwise. All the possibilities are summarized under the compact
formula:
1 k
1
q
kq j n lm
(2.34)
j l (g) mn
(g −1 ) =
nG g∈G
dk
where kq stands for a generalized Kronecker symbol, defined as kq = 1 if k ∼ q and kq = 0 if
k ≁ q . j n (resp. lm ) is the standard Kronecker symbol j n = 1 (resp. lm = 1) iff j = n (resp.
q
l = m) and 0 otherwise. If the matrix representations are unitary then mn (g −1 ) = ((q (g))−1 )mn =
q
q
†
⋆
(( (g)) )mn = nm (g) , which leads to the alternative formula:
1 k
1
q
(g) nm
(g)⋆ =
kq j n lm
nG g∈G j l
dk
(2.35)
The theorem can be proved also by directly using any pairof irreducible matrix representations k and
q and applying the Schur’s Lemmas to the matrix A = g∈G k (g) q (g −1 ), where is a dk × dq
matrix with entries all null except at line l and column m where it is set to lm = 1. The theorem is
straightforwardly extended to the finite-dimensional linear representations of compact groups
G on the
vector spaces over the field C. It suffices in the proof to replace every normalized sum n1G g∈G . . . over
a finite group G by the corresponding integration G . . . dg using the Haar measure dg of the compact
group G. It also is extended to every ground field
K whose characteristic char(K) that does not divide
q
k
(g) mn (g −1 ) can fail to give d1k if K is not
the order nG of the group G, except only that n1G g∈G nm
algebraically closed. This can be determined from the Galois Theory of the centre of the division algebra
EndG (V, K).
3. CHARACTER THEORY
What now one needs are effective methods for reducing a linear representation and constructing the
irreducible components of its representation space, to allow discerning the invariances of a physical
quantity with respect to a symmetry group. It is obvious from the considerations of the previous sections
that, quite quickly, this might become cumbersome. Invariants over the isomorphism classes of the
linear representations should be of the greatest help, at the condition that these also allow distinguishing
between non isomorphic linear representations.
Whatever the finite dimensional linear representation : G → GL(V, C) of a compact group G the
linear operators (g) for every element g in the group G are diagonalizable, since is unitarisable
and unitary operators are diagonalisable with pairwise orthogonal eigenspaces (cf. spectral theorem
for normal operators). It is recalled that finite groups are compact, for the discrete topology. As
a matter of fact, with finite groups it even may be asserted that all the eigenvalues of (g) are
roots of unity, since every element g ∈ G necessarily is of finite order, that is to say ∃ng : g ng = e
so that (g)ng = 1V . Numerical invariants may be deduced from the symmetric functions of these
eigenvalues, more precisely
from the coefficients n (g) of the characteristic polynomial Det[(g) −
1V ] = (−1)d d + dn=1 n (g)d−n , where d is the dimension of the representation space V. Among
the most familiar are the coefficient d (g) = Det[(g)] of the constant term and the coefficient
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Contribution of Symmetries in Condensed Matter
1 (g) = (−1)d−1 Tr[(g)] of the sub-leading term. It is clear that Det[ ◦ (g) ◦ −1 ] = Det[(g)]16
and Tr[ ◦ (g) ◦ −1 ] = Tr[(g)]17 whatever the invertible linear operator on V. Thus, Det[(g)] and
Tr[(g)] show the required invariance over every isomorphism class of linear representations. Now, it
follows from the multiplicativity of the Determinant that Det[(g) ◦ (h)] = Det[(g)]Det[(h)], which
means that the application g ∈ G → Det[(g)] ∈ C makes up a 1-dimensional representation of G. It
thus turns out that the Determinant invariant is often unable to distinguish between different classes
of isomorphism when, by contrast, the Trace invariant, which is not multiplicative, can. So this is the
searched invariant. It actually will be shown below that the complex-valued function on G defined as
: G → C, g → (g) = Tr[(g)]
(3.1)
is a complete invariant, in the sense that it uniquely determines the linear representation : G →
GL(V, C) up to isomorphism. defines the character of the linear representation .
3.1 Elementary properties
Let : G → GL(V, C) be a d-dimensional linear representation of a finite (or even continuous compact)
group G and let : G → GL(d, C) be the matrix representation associated to with respect to the basis
vectors eˆ m (m = 1, . . . , d) selected in the representation space V. It follows from the definition of the
trace of a linear operator that
ii (g) ∀g ∈ G
(3.2)
(g) = Tr (g) = Tr [(g)] =
i
It is the usage to also call the character of the matrix representation . The trace of a product of
matrices being invariant by cyclic permutation, we have ∀g ∈ G Tr[S (g) S−1 ] = Tr[(g)], whatever
the invertible matrix S. Of course, this is nothing but the transposition to the matrix representations of
the group G of the invariance of the character over an isomorphism class. concretely is independent
of any choice of basis vectors in the representation space V.
• (e) = dd, where e is the unit element of G. (e) = Tr [(e)] = Tr [Id ] = di=1 1 = d, where Id is the
d × d unit matrix.
• (g −1 ) = (g)⋆ and | (g) | ≤ d ∀g in every finite group G . ∀g ∈ G ∃ng ∈ N : g ng = e (unit element
of G), otherwise the successive powers of g would generate an infinite group. It follows that
(g ng ) = (g)ng = 1d . It then is directly clear that (g) is diagonisable. Let ǫ1 (g), . . . , ǫd (g)
ng
be the g-dependent eigenvalues of (g). Obviously,
means thatǫi (g) is a
√ ǫi (g) = 1, which
⋆
j i (g)
⋆
root
of
unity,
∃
(g)
:
ǫ
(g)
=
e
with
j
=
−1.
Now,
(g)
=
Tr
= i ǫi (g)⋆ =
[(g)]
i −1 i
−1
−1
−1
= Tr (g ) = (g ) whereas | (g) | = | Tr [(g)] | = | i ǫi (g) | ≤
i ǫi (g) = Tr
(g)
|
ǫ
(g)
|
=
1
=
d.
Note
that by the theorem of Lagrange the order ng of g divides the order
i
i
i
nG of the group G. So the eigenvalues ǫ1 (g), . . . , ǫd (g) of (g) are roots of unity of orders dividing
the order nG of the group G. More generally, every linear representation of a compact group and à
fortiori of a finite group is unitarisable. An inner product thus may be defined in the representation
space V so that (g −1 ) = (g)−1 = (g)† ∀g ∈ G. In terms of matrix representations with respect to
16 A Determinant most generally designates every alternating d-linear form F: End(M, A) → A on the module End(M, A) of the
endomorphisms on a free module M of dimension d over a commutative ring A. F is unique up to the image F(1M ) of the identity
endomorphism 1M . One standardly put F()/F(1M ) = Det[]. It results from the functorial properties of the exterior algebra on
the module M that Det is multiplicative: Det[ ◦ ] = Det[]Det[] ∀(, ) ∈ End(M, A)2 . As an obvious consequence, the image
by Det of any composition of endomorphisms i is invariant by any permutation of these: Det[i i ] = Det[i (i) ].
17 A Trace most generally designates every linear form F: End(M, A) → A on the module End(M, A), of the endomorphisms
on a free module M of dimension d over a commutative ring A, enjoying the property F( ◦ ) = F( ◦ ) ∀(, ) ∈ End(M, A)2 .
F is unique up to the image F(1M ) of the identity endomorphism 1M . One standardly put F()/F(1M ) = Tr[]/d. Obviously, by
substituting ◦ for and so on, the property F( ◦ ) = F( ◦ ) implies that the Trace of any composition of endomorphisms
is invariant under cyclic permutation, whence Tr[ ◦ ◦ −1 ] = Tr[] for invertible linear operators on a vector space. Note that
Det[e ] = eTr[] .
00005-p.19
EPJ Web of Conferences
an orthonormal basis, this transposes to (g −1 ) = (g)† = t (g)⋆ ∀g ∈ G (cf. Sections 2.5 and 2.6),
whence (g −1 ) = Tr[(g −1 )] = Tr[t (g)⋆ ] = (g)⋆ ∀g ∈ G.
• If # is the character of the representation # dual to the linear representation with character then
# (g) = (g −1 ) ∀g ∈ G
G. # (g) indeed acts on every linear form on V as the composition with (g −1 )
#
#
#
#
#
−1
: ∀v ∈ V = Hom(V,C),
v ◦i (g) . i
i (g)(v ) =
• The character of = i is = i , where stands for the character of i . Evident from the
property Tr [A ⊕ B] = Tr [A] + Tr [B] for any pair of matrices A and B.
3.2 Orthogonality theorem
setting j = l and n = m then summing over all j and all n and
Getting back to the equation
2.34 and finally using the identity j n (j n )2 = j n (j n ) = dk , one ends up at
1 q
( (g))⋆ k (g) = q | k = kq
nG g∈G
(3.3)
where k and q are the characters of the irreducible representations k : G → GL(Vk , C) and q :
G → GL(Vq , C). kq is a generalized Kronecker symbol, defined as kq = 1
if k ∼ q and kq = 0
1
k
q
| is used to emphasize that the quantity nG g∈G ((g))⋆ (g) does
if ≁ . The notation
define an inner product in the vector space C [G] of complex-valued functions on G, being obviously
linear with respect to , conjugate symmetric and positive definite ( | > 0 ∀ ∈ C [G] − {0}).18
Equation (3.3) makes up the First Orthogonality Theorem for the Characters and has far-reaching
consequences.
Consider a decomposition = 1 ⊕ . . . ⊕ s of a linear representation : G → GL(V, C) with
character into the irreducible representations k : G → GL(Vk , C) with characters k . It results from
the additivity property of the characters
that = 1 + . . . + s and from the linearity of the inner
product that q | = q | 1 + . . . + q | s . According to the First Orthogonality Theorem for
the Characters,
q ∼ k ⇐⇒ q | k = 1
q ≁ k ⇐⇒ q | k = 0
It follows that q | determines the number of k isomorphic to q contained in the decomposition
of . As previously transcribed in the equation (2.29), this number is nothing but the multiplicity nq of
q in the expansion of the representation over its irreducible components k :
∼
nk k =⇒ nq = q | (3.4)
k
The multiplicity of the trivial representation in this expansion for instance is g∈G (g). Obviously
nq = q | does not depend on the chosen decomposition, which means that the decomposition of a
finite-dimensional linear representation of a finite group into irreducible representations is unique.
This in turn immediately implies that every two completely reducible linear representations with the
same character are necessarily isomorphic, for they contain each given irreducible representation the
same number of times. Characters thus are in one-to-one correspondence with isomorphic classes of
linear representations, which is the essence of the Theorem of Complete Invariance of the Characters.
q
⋆ k
q
k
18 Of course, this may be extended to compact groups as
G ( (g)) (g)dg = | = kq by using the Haar integration
and to every ground field K whose characteristic char(K) that does not divide the order nG of the group G, with the proviso to
keeping in mind that the square norm q | q can fail to give 1 if K is not algebraically closed, that is to say we always have
orthogonality but not necessarily orthonormality.
00005-p.20
Contribution of Symmetries in Condensed Matter
Given that every decomposition
of a linear representation uniquely writes ∼ k nk k every
character uniquely writes = k nk k . Computing the square norm of and taking account of the First
Orthogonality Theorem for the Characters one gets
2
| =
nq q |
nk k =
nq nk q | k =
(3.5)
nk
q
k
qk
k
2
k nk
is equal to 1 only if one of the nk ’s is equal to 1 and the others to 0, that is if is isomorphic to
one of the irreducible representation k , whence if is the character of a representation then | is the sum of squares of integers and | = 1 iff is irreducible. We obtain thus a very convenient
irreducibility criterion.
3.3 Dimensional closure
Consider the regular representation G of a finite group G (cf. Section 2.1). G by definition transcribes
the left action of the group G on the representation space VG spanned by basis vectors eˆ h indexed
with the group elements h ∈ G by permuting these as G (g)(ˆeh ) = eˆ gh ∀g ∈ G ∀h ∈ G. It is clear
by the group properties that gh = h ⇔ g = e, where e is the unit element of G. It follows that
G (g)(ˆeh ) = eˆ h ⇔ g = e. This means that the diagonal elements of the matrix representatives G (g)
of the linear operators G (g) with respect to the basis {ˆeh }h∈G are all null for g = e and all equal to 1 for
g = e. The character G of of the regular representation G then is given by the formula:
nG if g = e
(3.6)
G (g) =
0 otherwise
where nG is the order of G. One finds that G | G = n1G g∈G (G (g))⋆ G (g) = n1G n2G = nG . So G
is far from being irreducible. If q stands for the character of an irreducible representation q : G →
GL(Vq , C) with dimension dq of the group G then one also computes
1 q
1 q
1
nq = q | G =
( (g))⋆ G (g) =
( (e))⋆ G (e) =
dq nG = dq
nG g∈G
nG
nG
It follows that
G =
dk k
(3.7)
k
that is to say the number of times each irreducible linear representation k is contained in the regular
representation G is equal to the dimension dk of that irreducible representation. The equation (3.7)
implies that G (g) = k dk k (g) for all g in G. Taking g = e leads to the dimensional closure identity
dk2 = nG
(3.8)
k
k
since G (e) = nG and (e) = dk . This identity is useful in the determination of the irreducible
representations of a group G, to check in particular that all of these have been found out. If g = e
then, since G (g = e) = 0,
dk k (g = e) = 0
(3.9)
k
Note that the span VG of {ˆeh }h∈G is isomorphic to the vector space C [G] of complex valued functions
on the group G. As to build an isomorphism it suffices to match the basis vector eˆ h in G with the
function h : G → C, g → gh . Under this isomorphism the elements g in G act on the left on C [G]
by sending the function to the function G (g)() such that G (g)()(h) = (g −1 h). As a matter
00005-p.21
EPJ Web of Conferences
of fact, this is the way to generalize the concept of regular representations to the compact groups.
The representation space VG then is isomorphic to the Hilbert space L2 (G, C) of the square integrable
functions on the group G and G (g) for each g ∈ G operates on this space by sending every ∈ L2 (G, C)
to G (g)() ∈ L2 (G, C) defined as G (g)()(h) = (g −1 h) ∀h ∈ G. It again is shown that the number
of times each irreducible linear representation k is contained in the regular representation G is equal
to the dimension dk of that irreducible representation, but now no dimensional closure prevails since the
group G is not finite. The regular representation G then is infinite-dimensional.
3.4 Class functions
Owing to the invariance Tr[ ◦ ◦ −1 ] = Tr [] of the Trace of any pair (, ) of invertible linear
operators on any vector space, the character of every linear representation : G → GL(V, C)
is conjugation-invariant:
(tgt −1 ) = Tr[(t) ◦ (g) ◦ (t −1 )] = Tr[(g)] = (g)
∀g ∈ G ∀h ∈ G
(3.10)
It is recalled that two elements g and h of a group G are conjugate iff there exists another element
t in the group G such that h = tgt −1 . Conjugacy is an equivalence relation that partitions the group
G into conjugacy classes Ci . A complex valued function on G is called a class function iff
(tgt −1 ) = (g) ∀g ∈ G ∀t ∈ G, that is to say iff it is constant over each conjugacy class Ci . It is
clear from the equation (3.10) that every character of a linear representation : G → GL(V, C) of
a finite group G is a class function.
The set of the class functions on a group G, endowed with addition and scalar multiplication makes
up a subspace C [CG ] of the vector space C [G] of the complex valued functions on G. Whatever the
linear representation : G → GL(V, C) of a finite group G and whatever the complex valued function
∈ C [G], we always may define a linear operator on V as:
(g)(g)
(3.11)
=
g∈G
is a class function iff commutes with the group G through any linear representation :19
∈ C [CG ] ⇐⇒ (h) ◦ = ◦ (h) ∀h ∈ G.
(3.12)
It follows that if is a class function and is isomorphic to an irreducible representation k : G →
GL(Vk , C) of the group G with character
k then, by Schur 1, ∃ ∈ C : = 1Vk (cf. Section 2.8). can be determined by computing Tr .20 As a partial conclusion, we write
nG ⋆ k | 1Vk
(3.13)
(g)k (g) =⇒ =
∈ C [CG ] and ∼
dk
g∈G
19
∈ C [CG ] ⇒ (h) ◦ ◦ (h−1 ) =
and
(h) ◦ ◦ (h−1 ) = ⇒
u∈G
g∈G
(g)(hgh−1 ) =
(h−1 uh)(u) =
u∈G
u(=hgh−1 )∈G
(h−1 uh)(u) = (u)(u) ⇒ (h−1 uh) = (u)
Note that the last deduction is obvious if we take for the regular representation G .
20

Tr = Tr 
g∈G

(g)k (g) =
g∈G
(g)Tr k (g) =
(g)k (g) = nG ⋆ | k and Tr 1Vk = dk .
g∈G
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Contribution of Symmetries in Condensed Matter
where nG is the order of G and dk the dimension of Vk . Now, assume that the class function
is orthogonal to thecharacter k of every irreducible representation k then, by virtue of the
equation (3.13), = g∈G (g)(g) is zero so long as is irreducible and by the decomposition
into irreducible representations we conclude that is always zero. Applying this to the regular
representation G and computing the image under of the basis vector eˆ e indexed with unit element e
of G, we obtain
(ˆee ) =
(g)G (g)(ˆee ) =
(g)(ˆeg )
(3.14)
g∈G
g∈G
but (ˆee ) = 0, since is zero, therefore (g) ∀g ∈ G, whence is the null function on G. In short
⋆ k
| = 0 ∀k =⇒ = 0
(3.15)
It is on the other hand clear from the equation (3.3) that the characters k of the irreducible
representations of the group G make up an orthonormal system in the space of the class functions
C [CG ]. In other words the characters of the irreducible representations of a finite group G form an
orthonormal basis for the space of the complex class functions C [CG]], which is the expression of
the Theorem of Character Completeness over the Class Functions. Again this is straightforwardly
generalized to the compact groups G by using the Haar integration for summation over G and
considering the Hilbert space L2 (CG , C) of the square integrable class functions on G. With the other
ground fields K the application of Schur 1 on the linear operator will involve the division algebra
End(Vk , K).
As an immediate consequence, the number of irreducible representations of a finite group G up
to isomorphism is equal to the number nC of conjugacy classes of G. Indeed, if C1 , . . . , CnC are the
distinct conjugacy classes of G then every class function ∈ C [CG ] is fully determined by its values
Ci ∈ C on each conjugacy class Ci . It therefore has nC degrees of freedom. This merely means that the
dimension of C [CG ] is nC , but, by the Character Completeness over the Class Function, this is equal
to the number of irreducible representations of G. This is still true of compact groups, but without any
interest since there then are infinitely many classes and infinitely many irreducible representations in the
group G.
Completeness
means that every class function ∈ C [CG ] on a group G is the linear combination
= k k | k of the characters k of the irreducible representations k of the group G. With the
class function g that takes the value 1 for every element of the class Cg = {h ∈ G | ∃ t ∈ G, h = tgt −1 }
n
and 0 elsewhere, we compute k | g = nCGg (k (g))⋆ where nCg is the number of elements in the class
Cg and nG the order of the group G. It follows, by definition of g , that
dC[C ]
1 if h ∈ Cg
nCg G k
⋆ k
( (g)) (h) = g (h) =
(3.16)
nG k=1
0 if h ∈
/ Cg
where dC[CG ] is the dimension of C [CG ], which, it is recalled, is equal to the number of classes nC in G.
Equation 3.16 makes up the Second Orthogonality Theorem for the Characters.
3.5 Character tables
Character Orthogonality, Complete Invariance and Completeness over the Class Functions offer the
great advantage to allow globally handling all the irreducible linear representations of a finite group
G up to isomorphism by means of the so-called Character Table. This is a square matrix with rows
labelled by the isomorphism classes of irreducible representations, columns labelled by the conjugacy
classes of the group and entries given by the values of the character for each isomorphism class of
irreducible representation and for each conjugacy class. Every linear representation of the group can be
characterized from this table by determining the multiplicities of its irreducible components from the
00005-p.23
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inner product with the rows of the table and even its decomposition into isotypical components from
projection operators on the representation space built over the irreducible characters as discussed in the
Section 3.6.
Given a finite group G the first stage to construct its Character Table is to find its conjugacy classes.
A series of properties of conjugate elements exist that ease this search. A few of them are:
⋆ The unit element e of every group always forms a conjugacy class {e} by its own.
⋆ In an abelian group every element form a conjugacy class by its own.
ng
⋆ The orders of the elements of the same conjugacy class Ci are all equal, since obviously gi i =
ng
ng
e and ∃t ∈ G : hi = tgi t −1 ⇒ hi i = (tgi t −1 )(tgi t −1 ) . . . (tgi t −1 ) = tgi i t −1 = e.
−1
−1
⋆ If hi is conjugate to gi then hi is conjugate to gi so that all the inverses of the elements of a given
−1
conjugacy class Ci belong to a same conjugacy class C−1
i . If gi and gi are conjugate then we a have a
−1
single conjugacy class, Ci = Ci , which is said ambivalent, otherwise we have two distinct conjugacy
classes Ci = C−1
i , which are said inverse of each other.
⋆ If nCi stands for the number of elements in each conjugacy class Ci then, inherently to the partition
of the group G into conjugacy classes, we have the class equation i nCi = nG where nG is the order
of the group G.
⋆ The elements of the conjugacy class Ci of any given element gi of the group G are in bijective
correspondence with the cosets of the normalizer NG (gi ) = {t ∈ G | tgi t −1 = gi }. NG (gi ) is a
subgroup of G so that G = e NG (gi ) + . . . + sj NG (gi ) + . . . + s[G:NG (gi )] NG (gi ), where [G : NG (gi )]
defines the index in G of NG (gi ). Conjugating gi with any element sj t of the coset sj NG (gi ) we
get (sj t)gi (sj t)−1 = sj tgi (t −1 sj−1 ) = sj gi sj−1 . On the other hand, if (sj t)gi (sj t)−1 = (sk r)gi (sk r)−1
then (sk r)−1 sj t gi (sj t)−1 sk r = gi so (sk r)−1 sj t = h ∈ NG (gi ) or else sj = sk (rht −1 ) which means
sj NG (gi ) = sk NG (gi ). It then is inferred that the conjugation of gi by the elements of distinct cosets
leads to distinct conjugates. Thus each conjugate of gi by an element of the coset sj NG (gi ) can be
j
uniquely labelled by this coset as gi . It follows that nCi is the index [G : NG (gi )] in G of the normalizer
of the representative gi of the conjugacy class Ci , but by the Lagrange Theorem [G : NG (gi )] =
nG /nNG (gi ) . Therefore nCi is a divisor of nG . It is recalled more generally that the normalizer NG (S) of
a subset S of elements of a group G is defined as NG (S) = {t ∈ G | tSt −1 = S}. A related concept is
the centralizer CG (S) of the subset S, which is defined as CG (S) = {t ∈ G | tS = St}. It goes without
saying that, obviously, the normalizer NG (gi ) of a single element gi of the group G is identical to the
centralizer CG (gi ) = {t ∈ G | tgi = gi t} of that element gi in the group G.
⋆ The intersection Z(G) = ∩g∈G CG (g) defines the Center of G. Z(G) is an abelian subgroup of G and
contains all the elements of the group G that form a class by their own.
..
.
The second stage to construct the Character Table of a finite group G is to get the list of the character
k of its irreducible linear representations k . In the case of small enough groups the already established
the equations
theorems
to find them all. We recall the elementary property k (e) = dk , 2 may be enough
k
d
=
n
and
d
(g
=
e)
=
0
inferred
from
the
regular
representation
=
G
G
k k
k k
k dk k , the
equality dC[CG ] = nC between the total number of the k and that of the conjugacy classes Ci and,
of course, the orthonormality of the k . Denoting ik the value of the character k of an irreducible
representation k : G → GL(Vk , C) over a conjugacy class Ci , the first orthonormality equation (3.3)
re-writes:
i
q
nCi (i )⋆ ik = nG kd
(3.17)
where nCi is the number of elements in the conjugacy class Ci and nG the order of the group G. This
makes up a “Row-by-Row Orthogonality Theorem” for the Character Table. The second orthonormality
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Contribution of Symmetries in Condensed Matter
equation (3.16) re-writes:
k
ik (jk )⋆ =
nG
ij
nCi
(3.18)
which makes up a “Column-by-Column Orthogonality Theorem” for the Character Table. It finally may
be remembered that, since G is a finite group, the character value ik is the sum of dk terms each of
which is an ngi -root of 1, the multiplicative unit of the complex numbers, where ngi is the order of the
elements gi of the class Ci .
Consider for purpose of illustration the geometric group of the rotations in the 3-dimensional space
about the center of a tetrahedron that leaves the tetrahedron invariant. It is denoted G = 23 by the
crystallographers and consists in 2-fold rotations about 3 distinct axes, that permute the summits by
pairs, and 3-fold rotations about 4 distinct axes, that permute three summits circularly. The group,
mathematically, is isomorphic to the group of even permutation of a set {a, b, c, d} of 4 objects. It is
recalled that a permutation is even iff it decomposes itself into an even number of transpositions,
that is to say iff its signature is sign() = +1. We have 3 elements of order 2: {gx ≡ (ab)(cd)},
{gy ≡ (ac)(bd)}, {gz ≡ (ad)(bc)} and 8 elements of order 3: {gt ≡ (abc)}, gt gx , . . ., {gt2 ≡ (acb)},
gt2 gx , . . .. With the unit e this corresponds to a group of order n23 = 12. One easily establishes that
gt gx gt−1 = gz , gt gz gt−1 = gy , gt gy gt−1 = gx and gt2 (gt gx )gt−2 = gx gt−2 = gt gy gt−1 = gt gy , . . ., which
leads to distinguish 4 conjugacy classes: C1 = {e}, C2 = {gx , gy , gz }, C3 = {gt , gt gx , gt gy , gt gz } and
C4 = {gt2 , gt2 gx , gt2 gy , gt2 gz }. We then must have 4 irreducible representations k (k = 1, 4) with
character k (k = 1, 4) and dimension dk (k = 1, 4). It is recalled that dk = 0 ∀k so the dimensional
2
closure equation
k dk = n23 imposes that d1 = d2 = d3 = 1 and d3 = 3. One of the irreducible
representations, 1 , necessarily is the trivial representation contained exactly once in the regular
representation, whence j1 = 1 (j = 1, 4), which fills the first row of the Character Table. Since
k (e) = dk , we have 1k = 1 (k = 1, 3) and 14 = 3, which fills the first column of the Character Table.
The other elements of the Character Table can be inferred from the orthogonality theorems for the
character, keeping in mind that 2k (k = 2, 3) are square roots ±1 of 1, 3k (k = 2, 3) and 4k (k = 2, 3) are
cubic roots {1, , ⋆ } of 1 with = exp ( 2i
), 24 the sum of d3 = 3 square roots of 1 and j4 (j = 3, 4)
3
the sum of d3 = 3 cubic roots of 1. Considering the C1 − C2 column-by-column orthogonality we get
1 + 22 + 23 + 324 = 0, with 22 = ±1 and 23 = ±1. 24 à priori can take the values −3, −1, 1, 3, to
which would correspond respectively the values 8, 2, −4, −10 for 22 + 23 . It follows that the only
consistent values are 22 = 1, 23 = 1 and 24 = −1, which fills the second column of the Character Table.
The C1 − C3 and C2 − C3 column-by-column orthogonality then imposes that 1 + 32 + 33 = 0 and
34 = 0. One similarly has 1 + 42 + 43 = 0 and 44 = 0 by the C1 − C4 and C2 − C4 column-by-column
orthogonality, which immediately fills the 4-th row. 1 + j2 + j3 = 0 (j = 3, 4) implies that if 32 = then 33 = ⋆ in which case 42 = ⋆ then 43 = by the C3 − C4 column-by-column orthogonality. We
finally get the Character Table:
Table 1. Character Table of the Tetrahedron Group 23.
1
2
3
4
C1 = {e} C2 = {gx , gy , gz } C3 = {gt , gt gx , gt gy , gt gz } C4 = {gt2 , gt2 gx , gt2 gy , gt2 gz }
1
1
1
1
⋆
)
)
1
1
= exp ( 2i
=
exp
( 4i
3
3
4i
⋆
)
1
1
= exp ( 3 )
= exp ( 2i
3
3
−1
0
0
The construction of the Character Table as above performed is rather unwieldy and reveals
itself inefficient as the order nG of the group G is increased. As a matter of fact, a number of
additional theorems may be formulated that offer tools to forge powerful search algorithms, taking
advantage of decompositions of groups into direct or semi-direct products of subgroups or else direct
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sums of subgroups, involving the concept of induced representation, making use of conjugacy class
multiplication, exploiting arithmetic properties of the characters, . . . . A few of these theorems and
methods will be approached in the following but only sketchily.
3.6 Projectors and exchangers
As to fully discern the effects of a symmetry group in the concrete instances it actually is inevitable to
have to explicitly determine the invariant subspaces of the linear representations. One then is sent back
to the discomforts of the arbitrariness associated with the intertwinings of the representations and of the
consequent lack in general of a natural decomposition of a completely reducible linear representation
: G → GL(V, C) of a group G into the irreducible representations k : G → GL(Vk , C). This clearly
prompts us to formulate a standard method, although not unique, of reduction.
An exception is the coarse-grained canonical decomposition = k k of the linear representation
into isotypical components k : G → GL(X
k , C), these being
isomorphic to the direct sum of nk copies
of the irreducible representations k : k = s k s and Xk = s Vk s with k s ∼ k and Vk s ∼
= Vk (s =
⊕nk
V
.
As
proved
from
Schur
2
the
canonical
1, nk ) or else, more symbolically, k ∼ nk k and Xk ∼
= k
decomposition is unique, which implies that the isotypical components k can be unambiguously
determined. k for each k is nothing but the restriction of to the representation space Xk and only
a little intuition is necessary to find out that each subspace Xk of the representation space V is fully
identified by the linear operator on V given by the formula
dk k
Pk =
( (g))⋆ (g)
(3.19)
nG g
It indeed is inferred from the equation (3.13) that the restriction of Pk on every subspace Vk s of V that
is isomorphic to the representation space Vk of the irreducible representation k is the identity operator
1Vk s ∼
=Vk and the zero operator on any other subspace of V. A linear operator the restriction of which on a
family of spaces is the identity (resp.
operator on the direct sum
zero) operator is the identity
(resp. zero)
s = 0
s (resp.
s ). It follows that P is the
space of the family, symbolically s 1Vk s = 1s Vk
0
k
s Vk
s Vk
identity operator on the representation space Xk = s Vk s of the isotypical component k and the
zero
operator everywhere else in the representation space V, that is to say Pk is the projector of V = q Xq
onto Xk .
Consequently, to formulate a method for a standard reduction of any linear representation of
a group G, it suffices to do so for each of its isotypical components k . Choose, in that purpose, a
basis {ˆen }n=1,...,dk in the representation space Vk of each irreducible representation k of G and denote
k : G → GL(dk , C) the matrix representation associated with k with respect to the selected basis in
each Vk . We are free to define for each k the linear operators
dk k
Qkmn =
(g −1 )(g) (m, n) ∈ {1, 2, · · · , dk }2
(3.20)
nG g nm
on the representation space V of . As from the orthogonality theorem for the matrix representations, to
k
be precise from the equation (2.34), it immediately
iss inferred that ∀(n, m) Qmn is null on every subspace
s
Vq=k and therefore on every subspace Xq=k = s Vq=k of V. One similarly establishes, focussing solely
at Xk , that if {ˆens }n=1,...,dk in Vk s stands for an isomorphic replica of {ˆen }n=1,...,dk in Vk then
s
eˆ m if n = r
Qkmn (ˆers ) =
(3.21)
0 otherwise
k
It thus is found out that Qkmm projects
representation space
the
Vm onto im(Qmm ) =
m
1
s
k
k
X
=
X
Span(ˆem , . . . , eˆ m , . . .) = Xk ⊂ Xk and that m Qmm = P so that k
m k . One also deduces
n
k
that Qkmn defines an isomorphism of Xm
k to Xk and is null elsewhere in the space V, that is to say Qmn
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Contribution of Symmetries in Condensed Matter
n
transforms Xm
through the equation (3.21) that Qkmn ◦ Qkrt = Qkmt if n = r
k into Xk . It further is shown
k
k
and zero otherwise and that (g) ◦ Qmr = n nm
Qknr . It follows that if xk1 ∈ X1k is a non null vector
m
m
k
1
then the vectors xk = Qm1 (
xk ) ∈ Xk are linearly independent and makes up a basis of a G-invariant
subspace Vk (
xk1 ) of dimension dk , which is isomorphic to Vk . Choosing a basis {
xk1 , . . . xks , . . .} in X1k ,
s
1
xk ), . . . Vk (
xk ), . . . the direct sum of which gives back Xk . It then is
one gets a collection of subspaces Vk (
clear that the restrictions of to these G-invariant subspaces can be taken as the k -copy components of
the searched standard decomposition of the isotypical component k . One may proceed systematically
in the concrete cases, by selecting an arbitrary basis in the representation space V of and projects each
k
k
vector of this basis onto the spaces Xm
k by using the projectors Qmm then applies the exchangers Qmn
to get the bases of all the standard G-invariant subspaces.
Generalization to the fields K whose characteristic char(K) does not divide the order nG of the group
G is straightforward as well as to the compact
and exchangers
groups G. In the latter case the projectors
are built by replacing the summation n1G g by the Haar integration: Pk = dk G (k (g))⋆ (g) dg and
k
Qkmn = dk G nm
(g −1 )(g) dg with (m, n) ∈ {1, 2, . . . , dk }2 .
4. MISCELLANEA
A few additional topics are more succinctly discussed in this section, in order to only catch a
glimpse of the wealth of the topic. Constructions of new linear representations of groups from existing
representations through tensor products of the representation spaces or through groups products are
described. The concept of induced representation is approached with a qualitative discussion of a few
essential theorems. A method of systematic search of the irreducible representations of finite groups
is mentioned. The section ends with a very short description of group representations on more general
mathematical objects than vector spaces.
4.1 Tensor product
A vector space V over a field K is the tensor product V1 ⊗ V2 of two vector spaces V1 and V2 over
the field K iff it is endowed with an application (v1 ∈ V1 , v2 ∈ V2 ) → v = v1 ⊗ v2 ∈ V linear in each
of the two variables v1 and v2 . It is shown that V is unique up to isomorphism. If {ˆeni i }ni =1,...,di (i = 1, 2)
is a basis of Vi (i = 1, 2) then {ˆen11 ⊗ eˆ n22 }n1 =1,...,d1 ,n2 =1,...,d2 makes up a basis of V: the dimension of
V1 ⊗ V2 is the product d = d1 d2 of the dimensions of V1 and V2 . The tensor product of vector spaces is
associative and distributive with respect to the direct sum, to be precise U ⊗ (V ⊗ W) ∼
= (U ⊗ V) ⊗ W
and (U ⊕ V) ⊗ W ∼
= (U ⊗ W) ⊕ (V ⊗ W) are natural isomorphisms. Natural is to mean that no choice
of basis is requested to produce the property. Let i (i = 1, 2) be a linear operator on the vector space
Vi (i = 1, 2). The tensor product 1 ⊗ 2 of the linear operators 1 and 2 is the linear operator
on the tensor product vector space V1 ⊗ V2 defined as (1 ⊗ 2 )(v1 ⊗ v2 ) = 1 (v1 ) ⊗ 2 (v2 ) ∀(v1 ∈
V1 , v2 ∈ V2 ). If Ai (i = 1, 2) is the matrix representative of i (i = 1, 2) with respect to the basis
{ˆeni i }ni =1,...,di (i = 1, 2) in the vector space Vi (i = 1, 2) then the matrix representative of 1 ⊗ 2 with
respect to the basis {ˆen11 ⊗ eˆ n22 }n1 =1,...,d1 ,n2 =1,...,d2 in the vector space V1 ⊗ V2 is the matrix A1 ⊗ A2 whose
entries are given in terms of the entries of the matrices Ai (i = 1, 2) as
(A1 ⊗ A2 )(i,k)(j ,l) = A1ij A2kl
(4.1)
which is checked by observing that the application of (A1 ⊗ A2 ) to the basis vector eˆ j1 ⊗ eˆ l2 contains the
basis vector eˆ i1 ⊗ eˆ k2 with the awaited coefficient A1ij A2kl . An interesting property is
Tr(A1 ⊗ A2 ) =
(A1 ⊗ A2 )(i,k)(i,k) =
A1i,i A2k,k =
A1i,i
A2k,k = Tr(A1 )Tr(A2 )
i,k
i,k
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i
k
(4.2)
EPJ Web of Conferences
If the operators i (i = 1, 2) are diagonalizable then so is A1 ⊗ A2 and if {ˆeni i }ni =1,...,di (i = 1, 2)
are the eigenbasis of i (i = 1, 2) with eigenvalues ni i (ni = 1, . . . , di ) (i = 1, 2) then so is {ˆen11 ⊗
eˆ n22 }n1 =1,...,d1 ,n2 =1,...,d2 with eigenvalues n11 n22 (n1 = 1, . . . , d1 , n2 = 1, . . . , d2 ). It then follows that
Det(A1 ⊗ A2 ) = {Det(A1 )}d2 {Det(A2 )}d1 .
Now let 1 : G → GL(V1 , C) and 2 : G → GL(V2 , C) be two linear representations of the group
G. The tensor product = 1 ⊗ 2 of the linear representations 1 and 2 is the linear representation
: G → GL(V, C) that associates to each g in G the linear operator (g) on the tensor product vector
space V = V1 ⊗ V2 such that (g)(v1 ⊗ v2 ) = 1 (g)(v1 ) ⊗ 2 (g)(v2 ), ∀v1 ∈ V1 ∀v2 ∈ V2 . is uniquely
defined up to isomorphism. The matrix representative (g) of the linear operator (g) for each g
in G with respect to the basis {ˆen11 ⊗ eˆ n22 }n1 =1,...,d1 ,n2 =1,...,d2 is the tensor product 1 (g) ⊗ 2 (g) of the
matrix representatives i (g) (i = 1, 2) of the linear operators i (g) (i = 1, 2) with respect to the bases
{ˆeni i }ni =1,...,di (i = 1, 2) in the vector spaces Vi (i = 1, 2):

 1
1
(g) 2 (g)
11 (g) 2 (g) · · · 1d
1


..
..
(g) = 1 (g) ⊗ 2 (g) ≡ 
(4.3)
 ∀g ∈ G
.
.
d11 1 (g) 2 (g) · · · d11 d1 (g) 2 (g)
One says that the matrix representation is the tensor product of the matrix representations 1
and 2 , symbolically = 1 ⊗ 2 . Generalization to multiple tensor product is obvious. Consider
then a linear representation : G → GL(V, C) of the group G. The ν-th tensor power of the vector
space V is the vector space V⊗ν = V ⊗ . . . ⊗ V (ν times) and the ν-th tensor power of the linear
representation is the linear representation ⊗ν : G → GL(V⊗ν , C) that associates to each g in G
the linear operator ⊗ν (g) = (g) ⊗ . . . ⊗ (g) (ν times) on V⊗ν . If {ˆen }n=1,...,d is a basis of V then
a basis in V⊗ν is obtained from the collection of vectors eˆ n1 ⊗ . . . ⊗ eˆ nν where the indices n1, . . . , nν
range over {1, . . . , d}ν : the dimension of ⊗ν is d ν . Applying ⊗ν (g) before or after any permutation :
eˆ n1 ⊗ . . . ⊗ eˆ nν → eˆ (n1) ⊗ . . . ⊗ eˆ (nν) of factors leads to the same result. This means that the action of
the group Sν of permutations commutes with ⊗ν . Sν thus must preserves the canonical decomposition
of ⊗ν . So every Sν -isotypical component of ⊗ν makes up a sub-representation of G. Among these
it is customary to discern the ν-th symmetric power Symν : G → GL(Symν V, C) associated with
the trivial representation of Sν and the ν-th alternate power Altν : G → GL(Altν V, C) associated
with the sign representation of Sν , which is defined by declaring that every transposition
produces
a multiplication by −1. Define the linear operators ± : v1 ⊗ . . . ⊗ vν → n!1 ∈Sν (±)N () v(1) ⊗
. . . ⊗ v(ν) on V⊗ν , where N () is the number of transposition under which decomposes. One
easily shows that + is a projector of V⊗ν onto Symν V and − a projector of V⊗ν onto Altν V.
The vectors + (ˆen1 ⊗ . . . ⊗ eˆ nν ) (1 ≤ n1 ≤ . . . nν ≤ d) make up a basis of Symν V and the vectors
− (ˆen1 ⊗ . . . ⊗ eˆ nν )(1 ≤ n1 < . . . nν < d) a basis of Altν V. If ν = 2 then one gets the symmetric
square Sym2 and the alternate square Alt2 . Note that ⊗ = Sym2 ⊕ Alt2 . The dimension of
Sym2 is dSym2 = d(d + 1)/2 and the dimension of Alt2 is dAlt2 = d(d − 1)/2. The matrix representation
associated with Sym2 with respect to the symmetrized basis {ˆen1 ⊗ eˆ n2 + eˆ n2 ⊗ eˆ n1 }1≤n1≤n2≤d defines the
symmetric square matrix representation [ ⊗ ] and the matrix representation associated with Alt2
with respect to the antisymmetrized basis {ˆen1 ⊗ eˆ n2 − eˆ n2 ⊗ eˆ n1 }1≤n1<n2≤d defines the antisymmetric
square matrix representation { ⊗ }. Of course ⊗ = [ ⊗ ] ⊕ { ⊗ }.
The characters of the tensor products!of linear representations are elementarily determined:
• The character of = i i is = i i , where i stands for the character of i . Evident from the
property Tr [A ⊗ B] = Tr [A] Tr [B] for any pair of matrices A and B.
• The character of the symmetric square [ ⊗ ] of with character is determined as
∀g ∈ G, S2 (g) = 12 (g)2 + (g 2 ) . Denoting ǫ1 , . . . , ǫd the eigenvalues of (g), one indeed com
putes S2 (g) = i≤j ǫi ǫj = i ǫi2 + i<j ǫi ǫj = i ǫi2 + 12 ( i ǫi )2 − i ǫi2 = 21 (( i ǫi )2 +
2
1
2
2
i ǫi ) = 2 (g) + (g ) .
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Contribution of Symmetries in Condensed Matter
• The character of the
antisymmetric
square { ⊗ } of with character is determined as
∀g ∈ G, A2 (g) = 12 (g)2 − (g 2 ) . Denoting ǫ1 , . . . , ǫd the eigenvalues of (g), one indeed
computes A2 (g) = i<j ǫi ǫj = 21 ( i ǫi )2 − i ǫi2 = 21 (g)2 − (g 2 ) . Note the equality 2 =
S2 + A2 , which reflects the fact that ⊗ = Sym2 ⊕ Alt2 .
..
.
A tensor product = k ⊗ q of two irreducible representations k and q of a group G generally
is not irreducible. Its decomposition into irreducible components t standardly writes
k ⊗ q ∼
nkq t t
(4.4)
t
t
where the multiplicity coefficients nkq are generically called Clebsh-Gordan coefficients. Using the
equations 3.3 and 3.4, these easily are computed as
1 t
( (g))⋆ k (g)q (g)
(4.5)
nkq t = t | k q =
nG g∈G
It immediately is inferred by comparison with equation (3.3) that if t (g) = 1 ∀g ∈ G then (k )⋆ = q .
In other words the trivial representation of a group G is contained once and only once in the reduction
of the tensor product k ⊗ q of any two irreducible representations k and q of G iff these are either
†
complex conjugate, k ∼ ⋆q , or adjoint of each other, k ∼ q .
4.2 Group products
The direct product G1 × G2 of two groups G1 and G2 by definition is the group formed by endowing
the set {(g1 , g2 ) | g1 ∈ G1 , g2 ∈ G2 } with the composition law
(g1 , g2 )(h1 , h2 ) = (g1 h1 , g2 h2 ) ∀g1 , h1 ∈ G1 and ∀g2 , h2 ∈ G2
(4.6)
If Gi (i = 1, 2) is of order nGi (i = 1, 2) then the order of G1 × G2 is nG1 ×G2 = nG1 nG2 . The group G1
is isomorphic to the subgroup G1 × E2 of the group G1 × G2 consisting in the pairs (g1 , e2 ) where e2 is
the unit element of G2 . It thus can be identified with it. The group G2 similarly can be identified with the
subgroup E1 × G2 of the group G1 × G2 consisting in the pairs (e1 , g2 ) where e1 is the unit element
of G1 . Each element of G1 × E2 obviously commutes with each element of E1 × G2 . Conversely,
let G be a group containing G1 and G2 as subgroups such that i- every g in G writes uniquely as
g = g1 g2 with g1 in G1 and g2 in G2 , ii- for g1 ∈ G1 and g2 ∈ G2 one has g1 g2 = g2 g1 . The product of
two elements g = g1 g2 and h = h1 h2 can then be written as gh = g1 g2 h1 h2 = (g1 h1 )(g2 h2 ). If we let
(g1 , g2 ) ∈ G1 × G2 correspond to the element g1 g2 ∈ G we then obtain an isomorphism of G1 × G2 onto
G. In this case, G is identified with G1 × G2 and one says that G is the direct product of its subgroups
G1 and G2 .
Now let i : Gi → GL(Vi , C) (i = 1, 2) be linear representations of the group Gi (i = 1, 2). We may
define a linear representation = 1 ⊗ 2 of the group product G1 × G2 on the tensor product vector
space V = V1 ⊗ V2 by setting
(1 ⊗ 2 )(g1 , g2 ) = 1 (g1 ) ⊗ 2 (g2 ) ∀g1 ∈ G1 ∀g2 ∈ G2
(4.7)
1 ⊗ 2 is unique up to isomorphism and is called the tensor product of the representations 1 and
2 . If i is the character of i (i = 1, 2) then the character of = 1 ⊗ 2 is given by (g1 , g2 ) =
q
1 (g1 )2 (g2 ) ∀g1 ∈ G 1 ∀g2 ∈ G2 . If k1 : G1 →
GL(V1 , C) and 2 : G 2 → GL(V2, C) are irreducible
q
q
q
q
q
q
representations then 1k | 1k = 1 and 2 | 2 = 1 so that 1k 2 | 1k 2 = 1k | 1k 2 | 2 = 1, which
q
means that k1 ⊗ 2 is an irreducible representation of G1 × G2 . Assume now that is a class
q
q
1k 2 , namely | 1k 2 =
function
on G1 × G2 , which is orthogonal to all the characters of the form
q
k
k
⋆ k
⋆
(g1 ,g2 ) ((g1 , g2 )) 1 (g1 )2 (g2 ) = 0. Fixing 1 and putting (g2 ) =
g1 (g1 , g2 )(1 (g2 )) we get
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⋆ q
2 (g2 )
q
= 0 ∀2 . is a class function so it is null. Since the same is true of every 1k
we conclude that is identically null on G1 × G2 . In order words, each irreducible representation of
q
G1 × G2 is isomorphic to a representation k1 ⊗ 2 , where k1 is an irreducible representation of G1 and
q
2 an irreducible representation of G2 . Obviously these properties allow completely reducing the study
of the representations of the group G1 × G2 to that of the representation of the groups Gi (i = 1, 2).
Given two groups G and H and a morphism of the group H into the group Aut(G) of the
automorphisms of G. The semi-direct product G ⋊ H of G and H with respect to the action of
H on G designates the set {(g, h) | g ∈ G, h ∈ H} endowed with the composition law
g2 ((g2 ))
(g, h)(g ′ , h′ ) = (g (h)(g ′ ), h h′ )
∀g, g ′ ∈ G and ∀h, h′ ∈ H
(4.8)
It is almost obvious that G ⋊ H shows a group structure. One has the so-called exact sequence
1 → G → G ⋊ H → H → 1, with the injective homomorphism I : G → G ⋊ H defined by I(g) =
(g, eH ), the surjective homomorphism S : G ⋊ H → H defined by S(g, h) = h and, as it is required
for an exact sequence, ker(S) = im(I). The subgroup I(G) = G × {eH } is normal. It is observed that
by identifying G with G × {eH } and H with {eG } × H every semi-direct product K can be conceived
as a semi-direct product of two subgroups G and H associated with the morphism of K into
Aut(G) defined by (h)(g) = hgh−1 with h ∈ H. Conversely if K is an extension of G by H, that is
if we have the sequence 1 → G → K → H → 1 with I : G → K injective, S : K → H surjective and
ker(S) = im(I) (exact sequence), and if K contains a subgroup L on which the restriction of S is an
isomorphism to H then K is isomorphic to the semi-direct product I(G) ⋊ L with the morphism of
K into Aut(G) defined by (h)(g) = hgh−1 with h ∈ H. K can be conceived also as isomorphic to a
semi-direct product M ⋊ L with each element k in K writing uniquely as gh with g in a subgroup M
isomorphic to G and h in a subgroup L isomorphic to H. It is clear that identifying a finite group as a
semi-direct product should be useful to the study of its (irreducible) representations.
Consider for purpose of illustration the Tetrahedron Group 23 discussed in Section 3.5. A simple
inspection of the composition of its elements suggests that it contains as subgroups the group M formed
by the set {e, gx , gy , gy } and the group L formed by the set {e, gt , gt2 }. M is normal but L is not.
M ∩ L = {e} and every element of 23 writes uniquely as gh with g ∈ M and h ∈ L. It follows that
23 = M ⋊ L. Note that 23 is not a direct product because M and L do not commute. Another reason
is that L is not normal. Now, the group L is isomorphic to the cyclic group C3 , which is abelian. The
characters i (i = 1, 3) of its irreducible representations, which are 1-dimensional, extends immediately
to 23 as i (gh) = i (h) (i = 1, 3) for g ∈ M and h ∈ L. The last character 4 of the group 23 then may
be obtaind by using for instance the row-by-row orthogonality theorem for the characters, to be precise
the equation (3.17).
When, more generally, a group G is the semi-direct product G = A ⋊ H of a group H with an abelian
group A then its irreducible representations are all obtained by the so-called method of “little groups”.
The characters of the irreducible representations of the abelian group A, which are 1-dimensional,
form a group X = Hom(A, C⋆ ). G acts on X by (g)(a) = (g −1 ag) for g ∈ G, ∈ X and a ∈ A (cf.
Section 4.5). Let (i )i∈X/H be a system of representatives for the orbits of H in X. For each i ∈ X/H,
let Hi be the subgroup of H consisting of those elements h s.t. hi = i and let Gi = A ⋊ Hi be the
corresponding subgroup of G. Extend the function i to Gi by setting i (ah) = i (a) for a ∈ A and
h ∈ Hi . Using the fact hi = i ∀h ∈ Hi one finds that i is a character of a 1-dimensional representation
of Gi . Let be an irreducible representation of Hi . Composing with the canonical projection Gi → Hi
gives an irreducible representation "
of Gi . By tensor product of i and "
an irreducible representation
of Gi is produced. Let i, be the corresponding induced representation of G (cf. Section 4.3).
i ⊗ "
It then is shown that i- i, is irreducible (by the Mackey criterion), ii- if i, ∼ i ′ ,′ then i = i ′ and
∼ ′ , iii- every irreducible representation of G is isomorphic to one of the i, .
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4.3 Induced representations
Let H be a subgroup of a group G and let g ∈ G. It is recalled that the set gH = {gh | h ∈ H} by
definition is the left coset modulo H containing the element g of G. Two elements h and f of G are
said congruent modulo H if they belong to the same left coset: ∀h ∈ G ∀f ∈ G {h ∈ gH ∧ f ∈ gH} ⇔
hf −1 ∈ H ⇔ h ≡ f (mod H ). Any two left cosets are either disjoint or identical. The set of left cosets
of H, denoted G/H, makes up a partition of G. If nG is the order of G and nH the order of H then the
number of elements of the set G/H defines the index [G : H] = nG /nH of H in G. If an element gL is
chosen in each distinct left coset then one gets a subset R = {g1 , g2 , . . . , g[G:H] } of G called a system of
representatives of G/H:
G = g1 H + g2 H + · · · + g[G:H] H
(4.9)
Each g in G writes uniquely as g = gL h, where h ∈ H and gL ∈ R is a coset representative.
Now let : G → GL(V, C) be a representation of the group G on the vector space V and
: H → GL(W, C) a representation of the subgroup H of G on a subspace W of V. If is a
subrepresentation of the restriction : H → GL(V, C), h → (h) = (h) of to the subgroup H then
the subspace (g)W depends only on the left coset gH. Indeed, if g is replaced by gh with h ∈ H then
(gh)W = (g) ◦ (h)W = (g) ◦ (h)W = (g) ◦ (h)W = (g)W, because W is H-invariant through
. We thus define a subspace W for each left coset in the set G/H, which is a replica of W in V.
By definition, the representation of the group G is induced by the subrepresentation of the
restriction of to the subgroup H of G iff
V=
W =
gL W
(4.10)
∈G/H
gL ∈R
It immediately is deduced that: i- If dW is the dimension of then the dimension of the induced
representation is dV = [G : H] dW . ii- If 1 is induced by 1 and if 2 is induced by 2 , then
1 ⊕ 2 is induced by 1 ⊕ 2 . iii- The regular representation G of G is induced by the regular
representation H of H. V then has a basis {ˆeg }g∈G indexed by G. So it suffices to take for W the
subspace of V spanned by the set of vectors indexed by the subgroup H of G that is {ˆeh }h∈H . iv- If
is induced by and if W1 is an H-invariant subspace of W then the subspace V1 = ⊕gLL∈R W1
is stable under G and the representation of G on V1 is induced by the representation of H in W1 .
Using some of these properties one proves the existence of the induced representations. We indeed may
assume, from property (ii), that is irreducible, in which case is isomorphic to a subrepresentation
of the regular representation H of H, which, according to property (iii), can be induced to the regular
representation G of G. Applying property (iv), we conclude that itself may be induced. Assume
that induces another representation ′ : G → GL(V′ , C). Whatever the linear operator : W → V′
we are free to extend it to the linear operator : V → V′ by putting = ′ (gL ) ◦ ◦ ((gL ))−1 on
each replica gL W of W. does not depend on the choice of the left coset representative gL and
is well defined since V is the direct sum of the replicas gL W. obviously is unique. Observe that
◦ (g) = ◦ (g) ∀g ∈ G. If is the injection of W into V′ then is the identity on W and satisfies
◦ (g) = ′ (g) ◦ ∀g ∈ G so that im() ⊃ ′ (g)W ∀g ∈ G whence im() ⊃ V′ . Since V′ and V
have the same dimension dV = [G : H] dW , we observe that is an isomorphism. As a consequence,
for every representation : H → GL(W, C) of a subgroup H of a group G on a subspace W of a
vector space V there exists a representation : G → GL(V, C) induced by , which is unique up to
isomorphism.
Choose a basis {ˆen }n=1,...,dW in W and let H be the matrix representation associated with the
en ) (n =
representation : H → GL(W, C) with respect to this basis. With the vectors eˆ
nL = (gL )(ˆ
1, . . . , dW ) a basis is built up in each replica gL W of W, whence, since V = gL ∈R gL W, the set
{ˆenL }n=1,...,dW ,L=1,...,[G:H] makes up a basis of V. ∀g ∈ G (g)(ˆenL ) = (ggL )(ˆen ) = (gM h)(ˆen ) since ggL ,
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as any element of G, necessarily belongs to
a unique left coset
gM H, but (gM h)(ˆen ) =
(gM )(h)(ˆen ) =
(gM )(h)(ˆen ) = (gM )(h)(ˆen ) = (gM ) m Hmn (ˆem ) = m Hmn (gM )(ˆem ) = m Hmn (ˆemM ). It
follows that the matrix representation H↑G associated with the induced representation with respect to
the basis {ˆenL }n=1,...,dW ,L=1,...,[G:H] is a matrix with the non zero-block component H ((gM )−1 ggL ) in the
(L, M) entry iff (gM )−1 ggL ∈ H and zero-block components in the other entries:
H↑G (g) =
(g, h) ⊗ H (h) with (g, h)M,L = (ggL , gM h)
(4.11)
h∈H
where stands for the Kronecker symbol: (ggL , gM h) = 1 iff ggL = gM h and 0 otherwise. H↑G is said
induced by H . The notation with an arrow H↑G is often evocative. We for instance have the transitivity
property: if H ⊂ G ⊂ M, W ⊂ V ⊂ U and : M → GL(U, C) is induced by : G → GL(V, C), itself
induced by : H → GL(W, C) then : M → GL(U, C) is induced by : H → GL(W, C). With arrows
this reads more concisely: (H↑G)↑M = H↑M .
The character H↑G of the representation induced by is straightforwardly obtained as
1 H↑G (g) = Tr(H↑G (g)) =
H (f −1 gf ) ∀g ∈ G
(4.12)
H ((gL )−1 ggL ) =
n
H
g ∈R
f ∈G
L
(gL )−1 ggL ∈H
f −1 gf ∈H
where H stands for the character of : H (h) = Tr(H (h)) ∀h ∈ H. The square of the character H↑G is
easily computed as:
1 (H↑G (g))⋆ H↑G (g)
H↑G | H↑G G =
nG g∈G
=
1 (H ((gL )−1 ggL ))⋆ H ((gL )−1 ggL )
nG g∈G g
L
1 +
(H ((gL )−1 ggL ))⋆ H ((gM )−1 ggM )
nG g∈G g =g
M
(4.13)
L
The first sum is simplified into n1G [G : H] nH H | H H = H | H H . It follows that H↑G is irreducible
iff H is irreducible and the second sum is null. If H is a normal subgroup of G then, defining the
conjugate HL of H by gL as HL (h) = H ((gL )−1 hgL ) ∀h ∈ H, one shows that the second sum is null
iff none of the conjugate HL of H by gL has common irreducible component with another distinct
conjugate HM of H by gM .
Let H ∈ C [CH ] be any class function on H. The complex valued function H↑G ∈ C [G] on G
defined by the formula
1 H (f −1 gf ) ∀g ∈ G
(4.14)
H↑G (g) =
nH f ∈G
f −1 gf ∈H
is said induced by H . Since it is a linear combination of characters, H↑G is a class function:
H↑G ∈ C [CG ]. Equation (4.14) merely extend the concept of induction to any class function. Consider
reciprocally G ∈ C [CG ] and denote G↓H the restriction of G to the subgroup H of G. This allows
formulating in a symmetric form the Frobenius Reciprocity Theorem:
H | G↓H H = H↑G | G G ∀H ∈ C [CH ] G ∈ C [CG ]
(4.15)
Equation (4.15) is useful is establishing the Mackey’s criterion of irreducibility of the induced
representations. Also required is the notion of double cosets: HgK = {hgk | h ∈ H, k ∈ K} for a
pair (H,K) of subgroups of G. These partition the group G into equivalence classes. Let S be a set
of representatives obtained with H = K, on choosing a single element in each distinct double coset.
00005-p.32
Contribution of Symmetries in Condensed Matter
A matrix representation Hs of the subgroup Hs = gs Hgs−1 ∩ H of H is defined for each gs in S by
putting Hs (h) = H (gs−1 hgs ) for h ∈ H. One then shows that H↑G is irreducible iff H is irreducible
/ H, that is have no common irreducible component. If H is a
and Hs and H↓Hs are disjoints ∀gs ∈
normal subgroup of G then Hs = H and H↑G is irreducible iff H is irreducible and not equivalent to
/ H.
any of its conjugate Hs by gs ∈
The concept of induced representations provide powerful tools to demonstrate a variety of important
theorems. We only mention among them the Artins’Theorem, which allows stating that each character
of a group G is a linear combination with rational coefficients of characters of representations
induced from cyclic subgroups of G. Induction is also extremely efficient in the determination of the
irreducible representations from representations of its subgroups. Note finally that the notion of induced
representations extends with the same definition to the compact groups G so long as H is a closed
subgroup of finite index. With infinite index the notion may be defined through the Hilbert space of
square integrable functions on the group.
4.4 Searching irreducibles
An essential problem of representation analysis is whether algorithmic procedures might be forged
that would allow finding out the invariant subspaces of any linear representation and the invariant
complements. A general method to determine the Character Table of any finite group can be given.
In that purpose let us consider back the conjugacy classes of a group.
We may define the “product” of two conjugacy classes Ci and Cj formally as the set Ci Cj =
{gi gj | gi ∈ Ci , gj ∈ Cj }. If g ∈ Ci Cj then any conjugate to g is also the product of an element of
Ci by an element of Cj , merely because hgi gj h−1 = hgi h−1 hgj h−1 . In other words, if an element of the
conjugacy class Cl appears a given number C(Ci Cj Cl ) of times in the set Ci Cj then every other element
of the same conjugacy class Cl will appear the same number C(Ci Cj Cl ) of times in the set Ci Cj . This
means that the conjugacy class product Ci Cj expands onto conjugacy classes Cl as
C(Ci Cj Cl ) Cl
(4.16)
Ci Cj =
l
where the class multiplication coefficients are strictly positive integers: C(Ci Cj Cl ) ∈ N − {0}. Ci Cj =
Cj Ci , since gi gj = gj (gj−1 gi gj ), so that C(Ci Cj Cl ) = C(Cj Ci Cl ). The expansion in the equation 4.16
contains the conjugacy class Cl = {e}, where e is the unit of the group G, iff the two conjugacy classes Ci
and Cj are inverse of each, merely because gi gj = e ⇔ gi = gj−1 , and whenever this is so the conjugacy
class e will appear nCi times in the conjugacy class product of Ci with itself if it is ambivalent and with
its inverse if this is distinct from it. In other words,
nCi if Cj = C−1
i
C(Ci Cj {e}) =
(4.17)
0 otherwise
Summing the linear operators k (g) over a class Ci the linear operator ki = gi ∈Ci k (gi ) is
defined on the representation space V. ki belongs to EndG (V )21 so, by Schur 1, ∃i ∈ C : ki = i 1Vk
(cf. Section 2.8), which in terms of characters is transcribed into nCi ik = i k (e). As from the
21
(h) ◦ ki ◦ (h)−1 =
g∈Ci
(h) ◦ k (g) ◦ (h)−1 =
g∈Ci
00005-p.33
k (hgh−1 ) =
u=hgh−1 ∈Ci
k (u) = ki .
EPJ Web of Conferences
equation 4.16 it is inferred that ki ◦ kj = gi ∈Ci k (gi ) ◦ gj ∈Cj k (gj ) = gi ∈Ci gj ∈Cj k (gi gj ) =
k
l C(Ci Cj Cl )
gl ∈Cl k (gl ) =
l C(Ci Cj Cl ) l , so i 1Vk j 1Vk =
l C(Ci Cj Cl ) l 1Vk whence
(nCi ik ) (nCj jk ) = k (e)
C(Ci Cj Cl ) (nCl lk )
(4.18)
l
If NC is the number of the conjugacy classes of the group G then this makes up a system of NC2 equations
over the NC variables ik (i = 1, NC ). This is the starting point of a variety of algorithms to determine
the Character Tables of the finite groups. Consult [6] for further details. The computations of irreducible
representations are harder, as emphasized in [7].
Arithmetic properties of the characters are also extremely useful. Note that since every element of
a finite group has finite order, the character values always are sums of eigenvalues that are roots of the
multiplicative unit, that is to say roots of a polynomial with coefficients in the set of integers Z. This
defines algebraic integers. It then follows, for instance, from the equation (3.13) that the dimensions
dk of the irreducible representations k : G → GL(Vk , C) are all divisors of the order nG of the group
G, since the set of algebraic integers is closed under addition and multiplication and since algebraic
integers given as rationals are in fact integers.
4.5 Group actions
Let : G → Aut(X) be a representation of a group G on a mathematical object X. One always
may define a function : G × X → X that canonically maps each couple (g, x) ∈ G × X into
(g, x) = (g)(x) ∈ X. It is straightforward to show that preserves the law of G, namely (gh, x) =
(g, (h, x)) ∀g, h ∈ G ∀x ∈ X, since is an homomorphism, and that the unit e of G is neutral for
, namely (e, x) = x ∀x ∈ X, because (e) necessarily is the identity of Aut(X). In other words, is
nothing but an action of the group G on the mathematical object X. Conversely, given an action
: G × X → X one always may define a function : G → Aut(X) that canonically maps each g ∈ G
into the isomorphism (g) : x → (g, x) of X. It is not more difficult to demonstrate that the properties
of an action imply that is a group homorphism. Accordingly, it is equivalent to define a representation
of a group G on a mathematical object X or an action of this group G on that object X. It then is
tempting to state that a representation is identical to an action, but that would make up a mathematical
abuse.
Using either of the two concepts of action or of representation, symmetry can be defined in a very
wide context. A subset Y of X is said invariant under a subgroup S of G if {(g, x) | (g, x) ∈ S × Y} ⊆ Y.
The elements of S then are called the symmetries of Y.
A group action : G × X → X is said isomorphic to a group action : G × Y → Y, symbolically
∼
, if they are intertwined with an isomorphism, namely if there exists an isomorphism : X → Y
which is equivariant: ((g, x)) = (g, (x)) ∀(g, x) ∈ G × X. Of course, if : G → Aut(X) and :
G → Aut(Y) are the representations canonically associated with and then ◦ (g) = (g) ◦ ∀g ∈
G that is ∼ .
The set Orb (x) = {(g, x) | g ∈ G} by definition is the orbit of x ∈ X. Writing xR y for y ∈
Orb (x) one gets an equivalence relation, which partition the set X into orbits. The quotient set defines
the orbit space X | G. If : G × X → X is an action of a finite group G on a manifold then X | G is
an orbifold with the singularities on the fixed points of in X. Interest in the orbifolds strongly raised
in the context of the geometrization conjecture, formulated by Thurston then proved by Perelman, as
essential pieces of manifold decompositions. An action is transitive if Orb (x) = X.
The set Stab (x) = {g ∈ G | (g, x) = x} by definition is the stabilizer of x ∈ X. It forms
a subgroup of G, whatever x in X. It is also called a little group. One easily establishes that
00005-p.34
Contribution of Symmetries in Condensed Matter
Stab ((g, x)) = g Stab (x)g −1 .22 It follows that the collection {Stab ((g, x)) | g ∈ G} of the stabilizers
of the elements of an orbit Orb (x) forms a conjugacy class of subgroups of G. If Stab (x) = G then
Orb (x) = x and x is termed a fixed point. If Stab (x) = {e} then Orb (x) is termed a principal orbit.
An action is effective if all its orbits are principal: Stab (x) = {e} ∀x ∈ X, which means that every
element of G other than the unit e of G acts by changing every element of X.
The function x : G/Stab (x) → Orb (x), from the set of the left cosets of the stabilizer Stab (x) in
G to the orbit Orb (x) is well defined and bijective. It then is inferred that: i- if G is finite then the number
of elements of any orbit with the same conjugacy class of stabilizers as Orb (x) is nOrb (x) = nG /nStab (x) ,
denoting nE the number of elements in a set E. ii- if is an infinitely differentiable action of a Lie
group then any orbit with the same conjugacy class of stabilizers as the orbit Orb (x) is a manifold of
dimension dOrb (x) = dG − dStab (x) . If dOrb (x) = dG − dStab (x) = 0 then the orbit is finite and its cardinal
is the quotient of the number of connected components of G over the number of connected components
of S.
A stratum by definition is the union of the orbits with the same conjugacy class of stabilizers.
An example is the set of the fixed points of the action. Another is the union of the principal orbits,
which consists in the points that are changed under any element of G other than the unit e of G. If is an
infinitely differentiable action of a compact group G on a real manifold X then every real valued function
invariant with respect to G possesses extrema on each stratum corresponding to maximal little groups,
namely proper little group not contained in any other proper little group, and all real valued function
invariant with respect to G have in common orbits of extrema, which precisely are those critical in their
stratum (consult [8]).
5. CONCLUSION
It is hoped that this little trip to the mathematical lands of linear representations of groups was not boring
in spite of the many digression made with respect to the initial scope of the lecture and that, instead,
was rather pleasant and enjoyable by providing an abstract glimpse of the basics on which the theory is
founded. The reported literature provides more details. Clearly, it by no way is exhaustive and emanates
only from the author’s own arbitrary taste.
References
[1] J.P. Serre, Représentations Linéaires des Groupes Finis (3rd Ed., Coll. Méthodes, Hermann, Paris
1978)
[2] M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Pub. Inc.
New York, 1989)
[3] Ya. G Berkovich and E.M. Zhmud’ , Characters of Finite Groups (Translations of Mathematical
Monographs Volumes 172 & 181, American Mathematical Society, 1998 & 1999)
[4] J. Stillwell, Naive Lie Theory (Springer Science, 2008)
[5] M. R. Sepanski, Compact Lie Groups (Graduate Texts in Mathematics, 235, Springer, 2000)
[6] J. D. Dixon, Numerische Mathematik 10, 446 (1967) and W. R. Unger, J. Symb. Comp. 41, 847
(2006)
[7] V. Dabbaghian-Abdoly, J. Symb. Comp. 39, 671 (2005)
[8] L. Michel, Rev. Mod. Phys. 52, 617 (1980) & L. Michel and B.I. Zhilinski, Phys. Rep. 341, 11
(2001)
22 Indeed, h ∈ Stab ((g, x)) ⇔ (h, (g, x)) = (g, x) ⇒ (g −1 hg, x) = (g −1 , (h, (g, x))) = (g −1 g, x) = x ⇒ ∃f ∈
Stab (x) s.t. h = gf g −1 . In addition, h = gf g −1 = h′ = gf ′ g −1 ⇔ f = f ′ .
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